Skip to main content

Full text of "Advanced exercises in practical physics"

See other formats


/ 



X' 



1/ 



Digitized by tine Internet Archive 

in 2007 with funding from 

IVIicrosoft Corporation 



http://www.archive.org/details/advancedexerciseOOschuuoft 



ADVANCED EXEECISES 



IN 



PEACTIOAL PHYSICS 



Uontion: C. J. CLAY and SONS, 

€AMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MARIA LANE. 

©lagflotn: 50, WELLINGTON STREET. 




ILeipjig: F. A. BROCKHAUS 

i^eto liorft: THE MACMILLAN COMPANY. 

aSombag: E. SEYMOUR HALE. 



^ADVANCED EXEECISES 

IN 

PRACTICAL PHYSICS 



BY 



ARTHUR SCHUSTER, Ph.D., F.R.S. 

LANOWORTHY PROFESSOR OF PHYSICS AND DIRECTOR OF 
THE PHYSICAL LABORATORIES IN THE OWENS COLLEGE, MANCHESTER, 

AND 

CHAELES H. LEES, D.Sc. 

LECTURER ON PHYSICS AND ASSISTANT DIRECTOR OF 
THE PHYSICAL LABORATORIES IN THE 0WKN8 COLLEGE, MANCHESTER. 




CAMBRIDGE : 

AT THE UNIVERSITY PRESS. 

1901 

[All Rights reserved] 



CambrilJgc : 

PRINTED BY J. AND C. F. CLAY, 
AT THE UNIVERSITY PRESS. 



PREFACE. 

nnHIS volume is intended for students who, having obtained 
-■- an elementary knowledge of experimental work in Physics, 
desire to become acquainted with the principles and methods 
of accurate measurement. The large and increasing number 
of students, who have to be taught simultaneously in a Physical 
Laboratory, renders it necessary that the instructions supplied 
should be fairly complete ; and that the exercises should be of 
such a nature as to enable the teachers easily to check the 
accuracy of the results obtained. The exercises described in 
this volume have been worked through by several hundred 
students of the Owens College who were preparing for the 
ordinary degree of B.Sc, and the experience thus gained has 
been utilised to improve the descriptions and methods adopted. 
It is hoped therefore that the volume will also prove useful in 
other laboratories. 

We have not aimed at completeness, being convinced that 
a student learns more by carefully working through a few 
selected and typical exercises, than by hurrying through ^ 
large number, which are often but slight modifications of 
each other. 

The guiding principle we have adopted in our teaching htis 
been to attach greater importance to neat and accurate work, 
properly recorded, than to the number of experiments which 
a student performs. All Note-books are carefully kept, no 
slovenly work is allowed to pass, and each exercise is repeated 
until satisfactory results have been obtained. 



VI PREFACE 

A student will naturally devote the greater portion of the 
time spent in the laboratory to measurements and quantitative 
work, but qualitative experiments should not be excluded. In 
certain parts of the subject, as for instance in Physical Optics, 
the educational value of setting up the apparatus and observing 
the general character of the effects produced is considerable, 
and such observations form a very useful complement to the 
quantitative exercises given in this book. 

We have endeavoured to confine the apparatus required to 
that commonly found in laboratories. It is not necessary that 
the instruments used should be identical with those described. 
Students should be able to introduce the slight modifications 
in the manipulation rendered necessary by some small differences 
in the apparatus. Where the differences are likely to be 
material, detailed descriptions have usually been omitted, and 
in such cases a written explanation should be supplied to 
the student with the instrument to be used. 



CONTENTS. 



BOOK I. 
PRELIMINARY. 

SECTION PAGE 

1 



T. Errors of Observation 

II. Meiisurement of Length 

III. Measurement of Intervals of Time 

IV. Calibration of a Spirit-Level 

V. Calibration of a Graduated Tube 



BOOK XL 
MECHANICS AND GENERAL PHYSICS. 



VI. The Balance 

VII. Accurate Weighing with the Balance 

VIII. Determination of the Density of a Solid . 
IX. Determination of the Density of a Liquid 

X. Moments of Inertia 



XI. E.xix;ri mental Determination of Moments of Inertia 71 

XII. The Compound Pendulum 

XIII. Experimental Determination of the Equivalent Simple 

Pendulum 

XIV. Determination of Young's Modulus by the Bending 

of Beams 

XV. Modulus of Rigidity 

XVI. Viscosity 

XVII. Surface Tension 93 



9 
12 
17 
22 



28 
40 
52 
62 
H7 



79 

82 

m 

89 



vm 



CONTENTS 



BOOK III. 
HEAT. 

SECTION PAGE 

XVIII. Coefficient of Expansion of a Solid .... 97 

XIX. Thennal Expansion of a Liquid .... 100 
XX. Coefficient of Increase of Pressure of a Gas with 

Temperatui'e 104 

XXI. Coefficient of Expansion of Air at Constant Pressure 108 

XXII. Effect of Pressure on the Boiling Point of a Liquid 112 

XXIII. Hygrometry 116 

XXIV. Laws of Cooling 119 

XXV. Cooling Correction in Calorimetric Measurements . 126 

XXVI. Specific Heat of Quartz 132 

XXVII. Latent Heat of Water 135 

XXVIII. Latent Heat of Steam 137 

XXIX. Heat of Solution of a Salt 140 

XXX. The Mechanical Equivalent of Heat . . . .142 

BOOK IV. 
SOUND. 

XXXI. Frequency of a Tuning-fork by the Syren . . 146 

XXXII. The Velocity of Sound in Air and other bodies by 

Kundt's method 148 

XXXIII. Study of Vibrations of Tuning-forks by means of 

Lissajous' figures 151 

BOOK V. 
LIGHT. 



XXXIV. Measurement of Angles by the Optical Method . 154 

XXXV. The Sextant 162 

XXXVI. Curvatures and Powers of Lenses . . . .167 

XXXVII. Determination of the Index of Refraction of a Liquid 

by Total Reflection 173 



CONTENTS 



IX 



SECTION PAGE 

XXXVIII. Magnifying Powers of Instruments .... 176 

XXXIX. Adjustments and Use of the Spectroscope . . 178 
XL. Reduction of Spectroscopic Measurements to an 

Absolute Scale 187 

XL I. The Spectrometer and its Adjustments . . . 194 
XLII. Determination of the Refractive Index of a Solid by 

the Spectrometer 203 

XLIII. Determination of the Index of Refraction of a Liquid. 

Specific Refractive Powers ..... 209 

XLIV. Photometry 213 

XLV. Interference of Light — the Biprism .... 216 
XLVI. Meivsurenient of Wave-length of Light by the Dif- 
fraction Grating 220 

XLVI I. Rotation of Plane of Polarisation. I . . . 224 

XLVIII. Rotation of Plane of Polarisation. II . . . 227 



BOOK VI. 

MAGNETISM AND ELECTRICITY. 

XLIX. The Measurement of Magnetic Fields . 232 

L. Magnetic Siu-vey of the Laboratory .... 239 

LI. The Tangent Galvanometer and other Current Meters 242 

LI I. Comparison of Current Meters 248 

LIII. Verification and Application of Ohm's Law 253 

LIV. Arrangement of Cells 268 

LV. The Water Voltameter 262 

LVI. Standardising Current Meters by the Copi)er N'oltji- 

meter 267 

LVII. On Mirror Galvanometers and their Adjustments . 272 
LVIII. Measurement of Resistance. The Post-Office Form 

of Wheatstone Bridge 286 

LIX. Measurement of a High Resistance .... 293 

LX. Measurement of a Low RoNi.^tance . . 298 

LXI. Meiisurcmont of the Resistjuico of a Galvan«»unt»r . :iiM» 

LXII. MeaMurcment of the Re-sistance of a Cell . 304 

LXIII. Measurement of Resistance by Carey Foster's Method ."107 



CONTENTS 



BBCnON 

LXIV. Change of Electrical Resistance with Temperature 

LXV. Resistance of Electrolytes .... 

LXVI. Construction of Clark Cell .... 

LXVII. Comparison of Electromotive Forces . 

LXVI II. Potentiometer Method of Measuring Currents 

LXIX. Thermo-electric Circuits .... 

LXX. Mea8iu*ement of the Mechanical Equivalent of 
by the Electric Method ... 

LXXI. Induction of Electric Currents . 

LXXII. Determination of the Inductance of a Coil 

LXX 1 1 1. Leakage and Absorption in Condensers . 

LXX IV. Comparison and Use of Condensers . 

LXXV. Determination of the Capacity of a Condenser 



Heat 



PAGE 
310 

315 
319 
323 

327 
329 

332 
335 
340 
350 
354 
356 



ERRATA. 



. 12, last line, for " minute " read " second." 

19, 3rd line from bottom, for "terms" read "turns." 

73, 20th line from top, after "distance apart." add "To do this place the 
vertical wire of the stand provided close to one edge of the block when it 
is in equilibrium. Set tbe block oscillating about a vertical axis and note 
the time required for say 30 or 40 passages of the edge of the block past 
the wire in the same direction." 

73, in table at foot of page multiply each number in the 6th and 7th columns 
by 10. 

74, line 11 from top, for " 7-5 " read " 7-45," and correct rest of line to 
correspond. 

74, line 12 from top, for "5*5" read "5-05," and correct rest of line to 

correspond. 
87, line 15 from top, after Section III. add " see correction to p. 73." 
141, in table at foot of page, for " 14-05 " read "16*05"; for " NHCl" read 

" NH^Cl." 



BOOK I. 

PRELIMINARY. 
SECTION I. 

ERRORS OF OBSERVATION. 

Our senses and judgment may be trusted up to certain 
limits, beyond which they begin to be subject to errors. Thus 
if we wish to measure a length of say 5 centimetres by means 
of a millimetre scale, no one will feel any difficulty in obtaining 
a result accurate to a millimetre or to half that quantity. But 
as soon as we wish to push the accuracy much further, even the 
most experienced observer will find the estimation difficult, and 
his measurement may be wrong by a quantity which is called 
an " error of observation." If he repeats the observations a great 
many times he will obtain a number of different results, which 
will group themselves round their average or mean value in a 
manner which will always shew a certain regularity, if the 
number of observations is sufficiently large. The study of the 
law of distribution of errors is of importance because it allows 
us to form an estimate of the accuracy with which under given 
conditions the measurements can be made. If there is no bias, 
which will cause the observation to err more often in one 
direction than in the other, common sense is sufficient to tell 
us that the arithmetical mean of a number of observations 
will give us the most probable result. And common sense will 

8. p. 1 



2 



PRELIMINARY 



also allow us to form a rough estimate as to the limits within 
which the result may be trusted to be right. Suppose for 
instance a certain observation three times repeated has given 
the numbers 31, 3*3, 34, and in another case the three ob- 
servations have been I'l, 1'5, 7*2. In both instances the most 
probable value, being the arithmetical mean, is the same, viz. 
327, but in the first case the observer may conclude with some 
confidence that his result is right to within ten per cent., i.e. 
the actual value will lie between 3 and 3'5, while in the second 
case he will attach little value to the mean obtained from such 
discordant measurements. 

Common sense like the sense of sight or of hearing may be 
trusted up to certain limits, and just as we can increase the 
efficiency of our ordinary senses by suitable instruments, so 
may we increase the efficiency of this common sense by an 
instrument which in this case is the theory of probability. 
To apply that theory we must in the first instance study the 
laws of grouping of errors, and this is best done by a graphical 
method. Let the curved line in Fig. 1 have the property that if 




N represents the number of observations supposed to be very 
large, the area EFHK will be a measure of the ratio njN, where 
n is the number of observations which shew an error greater 
than OH and smaller than OK. It follows of course that the 
unit of length chosen is such that the total area included 
between the curve and the line PQ is unity. 

It is found that in all cases which it is necessary to consider 
here, the curve has the same shape and may be represented 
analytically by the equation 

h 



Vtt 



r-h^X"^ 



I ERRORS OF OBSERVATION 3 

where x is the " eiTor," i.e. the deviation of an observation from 
the arithmetical mean. It is seen that different cases can only 
differ owing to a difference in the value of h, and Fig. 2 gives 




Fig. 2. 

the curves for three different values of h. Inspection of these 
curves shews that the greater the value of h, the steeper the 
curve in the neighbourhood of the central ordinate OAy and 
this means that the observations are grouped more closely 
round their average value. We might therefore take h to be 
the measure of the precision of our observations, but it is usual 
to choose for this purpose another quantity which we proceed 
to define. In Fig. 1 draw two lines LM and L'M' at equal 
distances from OA such that the area included between these 
lines the curve and the horizontal axis is equal to half unit 
area. By the definition of the curve this means that half the 
total observations shew errors numerically smaller than OL. 
The quantity OL is called the " probable " error, meaning that 
errors larger and smaller than that quantity are equally probable. 
The probable en-or (r) may be calculated from the equation to 
the curves in terms of h and is found to be given by 

hr = -4769. 

The probable error varies therefore inversely as h, and the 
smaller the probable error the more confidence may we have 
in our result. 

The quantity which interests us most, however, is not the 
probable eiror of an observation but the probable eiTor of the 
result which is obtained by taking the mean of all the observa- 
tions. Assuming the curve which has been given to represent 

1—2 



4 PRELIMINARY I 

correctly the distribution of errors, the probable error of the 
mean of N observations can be proved to be r/^N if r is the 
probable error of a single observation. This result is impor- 
tant, for it shews that by taking the mean of four observations 
we double the accuracy. It is seen that we may by repeating 
the observation easily double or even treble the reliability of 
the result, but that a very large number of observations have 
to be taken if we aim at a materially greater improvement. 

The curve which represents the distribution of errors in a 
series of figures which do not represent single observations, 
but each of which already represents the mean of a certain 
number of them, has the same form as that already described 
for the single observations. The probable error of the final 
mean again represents an error such that smaller and greater 
errors are equally probable. By finding the total area which 
lies on the right-hand side of such an ordinate as FK (Fig. 1) 
we find the probability of an error greater than OK, or in other 
words, we find the ratio n/N, where N is the total number 
and n the number of errors greater than OK. It may be 
proved in this way that an error equal or greater than 2r, i.e. 
double the probable error, will happen on the average in one 
out of every five cases, and may therefore be said to occur fre- 
quently. An error exceeding 3r will only occur once in 23 
cases on the average, while an error of or or more will only 
occur once in 1300 cases. We may say that we have a reason- 
able security though no absolute certainty that the result will 
not be affected by an error greater than four times the magni- 
tude of the probable error, the chance of greater error being in 
that case less than one per cent. It follows that one must take 
the mean of 16 observations in order to be reasonably certain 
that the error of the result shall not exceed the probable error 
of a single observation. 

The probable error for a given set of measurements might 
be obtained by dividing all the observations into two groups, 
according to the magnitude of the error. If the first group 
contains all observations with errors smaller than r and if by 
trial r is determined so that the two groups contain an equal 
number of observations, r is the probable error. But this 



I ERRORS OF OBSERVATION 5 

method can only be applied if a very large number of ob- 
servations is available. Having again recourse to the theory 
of probability the following method of calculation is found to 
give the correct result when the number of observations is 
large, and will in all cases give the best result obtainable. 

L.et Vi, Va...v„' represent the errors, i.e. the difference between 
the individual observations and their mean value. Then if 

the probable error of a single observation is given by 
and the probable error of their mean is 



Vn \ n.n — 1 



n being the number of observations. 

Sometimes the quantity to be determined by an experiment 
is not that which is directly measured, but is deduced by calcu- 
lation from the measurement. We must then be clear as to 
the error produced in the result by a certain error in the 
measurement. A simple case of this kind occurs if we wish 
to determine the area of a circle by measuring its diameter. 
If the diameter is D the area A is known to be JttZ^, but if an 
error of observation d has been committed so that the measured 
diameter was found to be Z) + d, the calculated area would be 
iir (D + dy or 

^ + a = i7ri>(l + 2j+g), 

where a is the error in the measured area. 

If d is small so that the scpiare of d/D may be neglected, 
we find 

A' D' 

If an error d has been committed in a measurement D, it 
is usual to call the quantity lOOd/D the percentage error. We 
may similarly call d/D the "per unit error" or better, the 



6 PRELIMINARY I 

"fractional error." The fractional error of the calculated area 
is therefore twice the fractional error of the measured diameter. 

More generally if a? be a measured quantity and y = ic*", the 
fractional error of y will be n times the fractional error of x. 
Hence the importance in such determinations as the coefficient 
of viscosity (Section XVI) of measuring as accurately as possible 
the radius of the tube, the fourth power of which enters into 
the calculation. 

As a further example we may take the measurement of an 
electric current by a tangent galvanometer. If the angle of 
deflection is 6 and the current C, the theory of the instrument 
gives the relation 

C = K tan 6, 

where k is the constant of the instru m ent. 

If an error 5 is committed in the measurement of 6, and c 
be the resulting error in the calculated value of G, we should 
have 

C + c = /c tan (^ H- 8) 

_ tan 6 + tan 8 
1 — tan 6 tan 8 * 

Since tan S is small we may neglect its square and sub- 
stitute h for its tangent, the above expression then transforms 
easily into 

C +c = K tan 6 + K^ sec^ 6, 

.'. c = kB sec" 0. 

The fractional error of the calculated current will therefore 
always be larger than twice the fractional error of the measured 
angle because sin 2^ will always be less than one. The error 
will be smallest when 6 = 4?5°, hence there is an advantage 
when measuring a current by a tangent galvanometer in select- 
ing the instrument so that the angle of deflection is nearly 45°. 

More difficult still are the cases in which there are two 
or more unknown quantities, connected by known relations. A 
simple case of this kind occurs when we wish to measure the 
time of vibration of a magnet, for instance by observing the 



I ERRORS OF OBSERVATION 7 

times of passage through the position of equilibrium. If t is 
the time of first passage and we only observe passages in the 
same direction, the successive recurrences of the event take 
place at times t, t-\-T, t + 2T, etc. The observed times are 
ti, to, U, &c., and if the errors are Vi, v^, v^, &c. we have the 
following equations 

ti + vo = T4-r, 

/3 + V3=T+2r, 
tn-^Vn = T + {n-\)T. 

These equations contain the two unknown quantities t and 
T and the n unknown values of the errors. There are therefore 
two more unknown quantities than equations. Any assumed 
value of T and T might be made to fit into the equations if 
there is no limitation to the magnitude of the error. The most 
probable of all possible values of r and T, are those which give 
that distribution of errors v which we have discussed and repre- 
sented by the curve in Fig. 1. The theory of probability shews 
that this is equivalent to saying that the most probable values 
of T and T are those which give the smallest values for S, where 

The method of calculation which allows us to determine the 
unknown quantities under the condition that S shall be a 
minimum is called the "Method of least squares." As there 
will be no occasion to reduce any of the observations described 
in this volume by this method, we need not enter into a fuller 
discussion, but students interested in the subject are referred 
to Merriman's Method of Least Squares. 

Errors of observation, which may be eliminated by taking 
a large number of measurements, are comparatively easy to deal 
with, but the experimentalist has also to guard against the 
more serious danger of being misled by "systematic errors" or 
errors which always affect the result in the same direction. 
These errors may be due to a faulty arrangement of the experi- 
ment, to defective instruments and also to a bias of judgment 
which causes the observer to commit errors which all tend to 



8 PRELIMINARY I 

lie on the same side of the correct value. If observations are 
subject to errors of the last kind they are said to be affected 
by a "personal equation." For instance, if a star is observed 
as it traverses the field of view of a transit instrument, and an 
observer is asked to make some sign at the moment the star 
passes behind a wire fixed in the instrument, it has been ascer- 
tained that most observers will signal the passage before it has 
actually occurred. With practised observers the difference in 
time between the actual and observed transit is always nearly 
the same, and is called the personal equation of the observer. 
Personal equations of small amount probably exist in many 
kinds of observation, such as the reading of a Vernier. It 
would follow that in such cases the symmetrical curve of Fig. 1 
only represents approximately the distribution of errors. But 
personal equations need only be taken into account in cases of 
extreme refinement of measurement and are only mentioned 
here in order to point out the existence of errors of observation 
which are not eliminated by the usual method of multiplying 
measurements. 



SECTION 11. 

MEASUREMENT OF LENGTH. 

The simplest method of measuring a length consists in 
the direct comparison of the length to be measured with a 
scale which serves as a standard, and which is subdivided into 
intervals of say 1 mm. If the length to be measured is not 
exactly a multiple of a millimetre, it becomes necessary in 
some way to measure or otherwise to estimate the fraction of 
a subdivision. 

The application of a " Vernier " to this purpose is explained 
in elementary books (see also Schuster and Lees's Intermediate 
Physics) ; the Vernier can however often be dispensed with and 
the subdivision estimated with sufficient accuracy by the eye. 

The following hints as to estimation will be found of use, 
but the student will only obtain confidence in his judgment by 
constant practice. He should not attempt to estimate more 
closely than to a tenth of a division, though an experienced 
observer will under favourable circumstances estimate correctly 
to a fiftieth. 

The eye subdivides easily an interval into two equal parts, 
and if the point to be measured is, e.g., at 
C (Fig. 3), between the scale divisions a C 3 

marked 2 and 3, etren an unpractised ob- p. 3 

server will put it down at once as 2*5. 
Should the point C lie as in Fig. 4 he 1 • ' ' ' 
will in his mind fix a point C at the 
same distance from C as that point is from '*^' 

the nearest scale division, then a point C" again at the same 



10 PRELIMINARY II 

distance from C. By this means he will see at once that G is 
less than a third but more than a fourth part of the way from 
the division 2 towards the division 3 ; hence he will read 
between 2'25 and 2*33 and there is no difficulty in putting 
it down as 2'3. 

In Fig. 5 an observer would see if he divided the interval 
between 2 and 3 into two equal parts at B, and . 
the first half again into equal parts at B\ that 2 CB* B 3 
the point G was rather nearer to 2 than to B 
and he would put it down as 2'2. ^^' 

The greatest liability to error occurs when the point G 
lies between '1 and '2 of an interval, and there will probably 
be a tendency to over-estimate the distance from the nearest 
division. 

In order to increase the power of judgment in this case, it 
is advisable for the student to draw two 
lines J. .and B and a third G, so that AG \ \ I 

is the tenth part of AB (Fig. 6). This ^^ . ^ 

should be done with AB varying in ^^' 

length between "5 and 3 cms. 

If a student once has got a good idea of a subdivision into 
ten parts, the estimation of subdivisions in general will not 
present much difficulty. 

Exercise. Estimate the distances of the middle lines from 
the left-hand lines in terms of the distance between the end 
lines taken as unity, in the cases shewn on the card provided. 
When these estimates have been entered in the note-book, 
measure the distances on a millimetre scale and check your 
estimates. 

When in the following exercises no special directions are 
given as to the method to be used in determining a length, 
a scale graduated in cms. and mms. should be applied to the 
length to be measured, adjusted to read an integral number of 
cms. at one end of the length, and the point on the scale corre- 
sponding to the other end, read. If this end coincides with a 
mm. division of the scale, the distance is read off immediately. 
If not, its position between two consecutive divisions must be 
estimated as explained above. 



II MEASUREMENT OF LENGTH 11 

When the length to be measured is small, a microscope 
provided with a divided transparent scale in the focal plane of 
the eye-piece is used. If, as for instance in the viscosity exercise 
(Section XVI), the radius of a narrow tube is to be measured, the 
tube is placed vertical under the microscope and the diameter 
read off directly in terms of divisions of the eye-piece. The 
value of the scale division must be determined by placing a 
transparent millimetre scale under the microscope, focussing 
and comparing it with the eye-piece scale. 

The divided scale of the eye-piece may be replaced by a 
" Micrometer," i.e. a cross wire which can be moved sideways 
by means of a fine screw the turns of which are read off on the 
screw head. There are several forms of micrometer which need 
not be described as the student will have no difficulty in using 
a micrometer eye-piece, should he meet with one fitted to an 
instrument. 



SECTION III. 

MEASUREMENT OF INTERVALS OF TIME. 

All time measurements in a Physical Laboratory are re- 
ferred to some standard clock, watch, or chronometer, the 
error of which must be known in the few instances in which 
absolute time is required. In the great majority of cases 
intervals of time only are measured, and the rate only of the 
clock is required. 

A clock or watch is generally available which goes correctly 
to within a minute a day, i.e. one minute in 1440. 

Such an instrument would therefore allow us to determine 
intervals of time to within less than one part in a thousand, 
an accuracy sufiBcient for many purposes. When greater accu- 
racy is aimed at, or when the error of the timepiece is too 
great, the rate must be determined by comparison with some 
better instrument or by direct observations of the sun or a star 
with a sextant or transit instrument. 

The method to be adopted in measuring an interval of time 
varies with the nature of the interval, but in most cases which 
occur in the laboratory the interval is that between two con- 
secutive occurrences of some periodic event, e.g. the passage of 
a pendulum through its position of equilibrium. The simplest 
course to pursue is then to count the number of occurrences 
in a given tii^ie, say one or two or more minutes, according to 
the accuracy required. The number of occurrences divided by 
the time elapsed between the first and last gives the required 
interval. 

If the occurrences follow each other rapidly, as e.g. when 
they take place four times a minute, it requires a little practice 



Ill 



MEASUREMENT OF TIME 



13 



to count them. This is most easily accomplished by taking 
them in groups of four, counting thus: one-two-three-four, 
one-two-three-four, etc., stress being laid on the four in pro- 
nouncing, and a mark being made simultaneously with a pencil 
on a sheet of paper. Four times the number of marks made 
on the paper in a given time is the number of occurrences 
duiing the time. 

If the peiiodic time exceeds ten seconds, the total time 
required for a given number of occurrences is great, and the 
accuracy of the determination of the interval may be increased 
without unnecessarily increasing the time occupied, by ob- 
serving separately the times of a number of occurrences. In 
this case a watch or clock beating seconds or half-seconds is 
required. The observer begins a few seconds before the occur- 
rence to count seconds in time with the clock, while he watches 
for the occurrence, the counting being done thus : — twenty-o?ie, 
twenty-^i^o, twenty -three, the twenty being spoken lightly, and 
stress being laid on the ttvOy three, etc., each of which should 
be pronounced in coincidence with the tick of the clock. By 
exercising a little care the twenty may be pronounced just at 
the half-second, and the exact time of the occurrence fixed to 
less than half a second. The times of consecutive occurrences 
numbered 1, 2, 3, 4, 5 etc. are found in this way, and written 
down. 

To shew how the observations are reduced we take as 
example the determination of a time of vibration, 12 successive 
passages for instance of a galvanometer needle through its 
position of equilibrium in the same direction being observed. 
Let the observed times be : — 



Event 
No. 


Time 


Event 
No. 


Time 


1 


lib 23- 8- • 


7 


ll'»24» 1' 


2 


12 


8 


11 


3 


22 


9 


20 


4 


32 


10 


30 


5 


41 


11 


39 


6 


51 


12 


49 



14 PRELIMINARY III 

If we only make use of the first and last observations we 
should find the interval of time for 11 vibrations to have been 
1" 46', and therefore the time of a single vibration to have 
been 9^-64. 

The intermediate observations are not made use of in this 
method of calculation, and there is therefore some loss of accu- 
racy in the final result. To discover the method of reduction 
which, without too much numerical labour, should give the 
best result, we notice in the first instance that the probable 
eiTor in the measurement of a time interval will be the same, 
whether that interval be large or small. If e is that probable 
error, and if the measured interval includes n complete vibra- 
tions, the probable error of the periodic time calculated from 
the n intervals will be ejn. 

If instead of observing the time of n vibrations we had 
observed the time of a single vibration, and repeated the ob- 
servation n times, the mean of the results so obtained would 
have given a probable error ej^Jn (Section I). As ejn is smaller 
than e/Vw in the ratio of Vn : 1, it follows that a better result 
is obtained if only two observations are taken, viz. one at the 
beginning and one at the end of n consecutive intervals, than 
if n separate intervals implying 2n observations are taken. 
This shews that it is not only the number of observations which 
determines the accuracy of a result, but their intelligent ar- 
rangement and reduction. Returning to the above example 
it is easily seen that the result may be improved by making 
use of the two first and two last observations. The interval 
between the first and the 11th is 96* for 10 vibrations or 9'60 
for one, the interval between the second and last is 97* for 
10 vibrations or 9-70 for one, the mean being 9*65. 

The probable error of each of these results separately 
is e/10 and that of their mean therefore e/10 V2 or nearly e/14, 
as compared with e/11 in case only the first and last obser- 
vations are made use of. 

More generally taking the first p and the last p observa- 
tions and taking the difference between the {n —p + l)th and 
the first, that between the {n—p-\- 2)th and the second, and 
so on until we come to the difference between the pih. and 



Ill 



MEASUREMENT OF TIME 



15 



last observation, we secure p values of n — p intervals, the time 
of vibration calculated from each separately will have a pro- 
bable error el{n—p) and the probable eiTor of their mean will 
be e/(n — p) y/p. The smallest probable error is obtained when 
(n —p) hjp is as large as possible for a given value of n, and it is 
easily shewn that this is the case when p — w/3. 

Appl}dng this to the above example it follows that it would 
be best to reject the 5th, 6th and 7th observations and to 
arrange the rest as follows: — 



Event 
No. 


Time 


Event 
No. 


Time 


Time of 8 
intervals 


1 
2 
3 
4 


11»>23'°3« 
12 
22 
32 


9 
10 
11 
12 


1 1^ 24°* 20" 
30 
39 
49 


jm 17. 

18 

17 
17 



Sum 


5- 9- 


Mean 


1 17-25 


.*. Interval 


9-656 



The observations which are not made use of need not be 
written down at all, but in that case great care is needed to 
avoid making an error in the number of observations which 
are omitted. The liability to error is reduced, and the re- 
liability of the result not materially interfered with by a 
slightly different arrangement. If 'p intervals are observed, 
i.e. jo + 1 observations taken, then p intervals omitted, and 
p again observed, the time when the last set of p observa- 
tions should commence may be readily calculated. Thus in 
the above example if four intervals had been measured the 
diflference of the first and fifth observation would have given 
38' for the time of four intervals, adding 38" to the fifth obser- 
vation the calculated time for the ninth would have been 
llh 23™ 41' -I- 38' = 11'* 24°* 19', the observer could therefore 
have rested till near that time and then carefully watched for 
the occurrence, which as the table shews actually took place 
at 24™ 20-. 



16 PRELIMINARY III 

The method which has been explained in connection with 
the measurement of time intervals, is equally applicable in 
other cases, such as that of temperatures measured with a uni- 
formly rising or falling thermometer read off at equal intervals 
of time, the rate of fall or rise of the thermometer being the 
quantity to be determined (Section XXIV). Sections (XIX) 
and (XX) also shew examples of the same method of reduction. 




SECTION IV. 

CALIBRATION OF A SPIRIT LEVEL. 

Apparatus required : Spirit Level, hoard with levelling 
screiv, screw gauge. 

The Spirit Level consists of a slightly bent tube (Fig. 7) 
partly filled with alcohol. The 
bubble of air or of vapour of 
alcohol left in the tube will al- 
ways set itself so that it is at 
the highest point of the tube. 
The tube is generally divided as 
shewn in Fig. 8, so that it is 
easily seen when the bubble is in 
the centre of the tube. If the 
level is in proper adjustment the 
base of the plate to which the level is attached should be hori- 
zontal when the bubble is in the central position. It is therefore 
necessary that the central point D of the tube (Fig. 7) should be 
farthest from the supporting plate AB. An adjusting screw T 
is generally provided on one side of the level, by means of which 
the tube can be tilted with respect to the base to secure that 
this shall be the case. If the table on which the level is placed 
is nearly but not quite horizontal, the bubble may not stand in 
the middle, but be slightly displaced ; the position of its centre 
may be read off in scale divisions, and if the value of the scale 
division is known, the inclination of the table to the horizontal 
may be calculated. It is the object of the present exercise to 
calibrate the divisions of a level so as to find their values in 



I I ft.i.|iiMHc =^|m.|iiio loj 



Figs. 7, 8. 



8. P. 



18 PRELIMINARY IV 

angular measure, and also to shew how even if the spirit level 
is not itself in proper adjustment it may be used to determine 
whether a surface is horizontal. 

If the level has not been tested and when placed on a table 
shews the centre of its bubble shifted to one side, it is un- 
certain whether this is due to the fact that the level is not in 
adjustment or that the table is not horizontal. But if the level 
be turned through 180° round a vertical axis so that the points 
A and B are interchanged, we may find out which is the correct 
interpretation. If the table is horizontal and the level wrong, 
the bubble should remain at the same place, for that point of 
the tube which is farthest from AB will remain so in whatever 
position the level is placed. If on the other hand the level is 
correct and the table inclined, the bubble will move over to 
the other side and its new position will be changed with 
reference to the level, but will be the same in space as 
previously. 

You are provided with a board supported on three legs, one 
of which can be screwed up or down so as to alter the angle of 
inclination of the board to the horizontal. In order to be able 
to measure the amount through which the board is tilted, 
the movable leg carries a divided circle in the manner of 
a* spherometer screw. The rotation of the screw is read 
on the divided head and its pitch must be separately de- 
termined. The board is to be placed with its legs on thick 



^ 



Fig. 9. 

glass plates, and the Spirit Level placed on the marked space 
at the centre of the board, with its length perpendicular to the 
line joining the fixed legs of the board. One end of the level is 
provided with an adjusting screw. 

Take a note of the side on which the adjusting screw is 



IV CALIBRATION OF A SPIRIT LEVEL 19 

placed. Let it be for the sake of uniformity the right-hand 
side. If the level has no adjusting screw, mark one end in 
some way and place it to the right. 

Level the board by means of its screw till the bubble is 
nearly in the centre. 

Let the readings of the ends of the bubble be, e.g., — *4 and 
4- '5, the divisions at the screw end of the level being called 
positive, those at the other end negative. 

Now remove the level and reverse it so that the screw 
end is to the left. If the reading of the screw end of the 
bubble remains 4- "5, the board is horizontal. If the reading 
is altered to + '6 say, take out half the error by means of the 
board screw, so that the end of the bubble now reads + '55. 

Reverse the level again so that its screw is on the right 
hand. 

This reversal should not alter the reading of the screw end 
of the bubble from 55. 

If it does, again take out half the error by means of the 
board screw, and reverse again. 

In this way a position of the board screw is found, such that 
reversal of the level produces no change in its readings. 

The board is then horizontal. 

Adjust the screw of the level till, when the level is on the 
board, the two ends of the bubble are at equal distances from 
the centre of the scale. 

The level is then in proper adjustment for use. 

If the level provided has no adjusting screw, the error of 
the level is thus determined and the calibration may be pro- 
ceeded with. 

To determine the value in degrees, minutes, and seconds, of 
each division of the level, place it with its screw end to the 
right. Adjust the board screw till the centre of the bubble is 
at 0, and read the screw head. 

Now, adjust the board screw till the centre of the bubble is 
at 1 to the right of zero, and read the screw head, recording the 
number of terms and decimals of a turn. Repeat, placing the 
centre of the bubble at 2, and so on for the other divisions of 
the right-hand scale. 

2—2 



20 



PRELIMINARY 



IV 



Again adjust the centre of the bubble to 0, read the screw 
head, and take observations as the bubble is moved to the left. 

Determine the pitch of the screw by first adjusting the 
screw till the centre of the bubble is at zero, reading the screw 
head, then inserting under the screw the small piece of plate- 
glass 5 mms. thick provided, again adjusting the screw till the 
bubble is at zero, then reading the head and recording the 
number of turns and parts necessary to make the adjustment. 
Determine by the screw gauge the thickness of the glass-plate, 
and thence calculate the pitch of the board screw. 

Measure the perpendicular distance / from the screw leg to 
the line joining the other two legs of the board. 

Then if n = number of turns of board screw from its reading 
when the bubble was at 0, and p = pitch of the screw, 

np = distance through which screw has been raised, 

and np/'l = angle in circular measure through which the board 
has been titled. (Since the angle is small.) 



Or in angular measure 



. 180° np 



TT 



Determine in this way the angle of tilt for each observation, 
and draw up a table as follows : — 

Spirit Level A . 



Reading of centre of bubble 


Board screw 


d 


5^ 


on screw end of scale 


head 





•100 








1 


1-350 


9' 2" 


9' 2" 


2 


2-650 


18' 25" 


9' 23" 


3 


3-783 


26' 36" 


8' 11" 


4 


4-567 


32' 11" 


5' 45" 


5 


6-183 


36' 42" 


4' 31" 


6 


5-683 


40' 19" 


3' 37" 



Similarly for readings of the bubble on the other end of 
the scale. 



IV CALIBRATION OF A SPIRIT LEVEL 21 

The column headed 86 gives the angle through which the 
level has to be turned in order to shift the bubble through one 
division. If the curvature of the tube of the level is cx>nstant, 
so that its axis is a circle, the numbers in the last column 
should be the same. The particular level to which the above 
numbers refer had a greater curvature nearer the end than at 
the centre. 

Students may find that the length of the bubble is not 
constant owing to the evaporation or condensation of the 
alcohol due to changes of temperature. Hence the necessity 
for reading both ends of the bubble. 



SECTION V. 

CALIBRATION OF A GRADUATED TUBE. 

Apparatus required : Graduated tube, mirror^ lens to he 
used as Tnagnifying -glass, and clean mercury. 

The graduations on a tube which is to be used for 
measurements, are as a rule placed at equal distances apart 
along the tube, but if the tube is used to measure volumes, 
the assumption that distances between the graduations repre- 
sent equal volumes of the tube, will lead to error unless the 
tube is of uniform bore. It is the object of calibration to find 
the correction which must be applied to the reading on the 
scale of equal lengths, in order to convert it into one of equal 
volumes. The calibration correction is that part of a division 
which must be added to any reading on the equal length scale 
in order to obtain the reading of the same point on an equal 
volume scale. 

Let a tube be provided with equidistant graduations, 1, 2, 
3, &c., see Fig. 10, and imagine graduations I., ii., ill., &c., along 
the same tube, such that the intervals between each represent 
not equal lengths, but equal volumes of the bore of the tube. 





1 


1 

1 


2 3 « 

1 1 1 


a 
1 


t 


1 




1 


1 1' 1 

D nr 11 
Fig. 10. 


1 

V 


1 



Thus the zeros of the scales being supposed to coincide, the 
division 1 corresponds to rather more than I. on the equal 
volume scale line, the correction at the point 1 is therefore 
positive. Similarly the corrections at points 2 and 3 are 
positive, and at 4 negative. 



V CALIBRATION OF A GRADUATED TUBE 23 

Let the corrections at successive divisions be Xq, a?,, x^, a?,, 
&c., and suppose a mercury thread, which if the temperature is 
constant will be of constant volume, say about equal to that of 
a scale division, to be pushed along the tube. 

In Fig. 10 the ends of the thread lie approximately at the 
points marked 3 and 4. Let 84* be the excess of the length of 
the thread over the distance between the points 3 and 4. The 
point 3 of the equal length scale would read 3 + .^3 on the equal 
volume scale, the point 4 would read 4s + x^, the true distance 
between these points would be therefore, not 1 but 1 +x^ — Xi, 
and if a be the number of divisions between the points 3 and 4 
on the equal length scale, the subdivisions being left out in the 
figure, the true length of the interval would be not a but 
o + a;4 — x^. 

The excess ^4' read on the equal length scale being small, 
will not be sensibly different when read on the equal volume 
scale, hence the coirected length of the thread becomes 
a 4- ^4 — a:, + 84' and this we know must be equal to a constant, 
say I, wherever the thread is placed. 

Applying this equation successively to the different positions 
of the thread we obtain the following n equations 

^'1 — Xf, -\- hi'' = I — a 
x^ — Xi'\'h^ = l~-a 



a?n-a:n_i + 5n*»-' = ^-a 



(1) 



involving n-\-2 unknown quantities, a?©. ^i,---^n, ^' Hence we 
must know two of the corrections, or at any rate two relations 
between them, to solve the equations. 

An important case of calibration is that of a thermometer 
between freezing point and boiling point, the corrections at 
these two points being determined by experiment, i.e., x^ and ar^ 
are known, and there only remain n unknown quantities. 

In that case, to solve the above set of equations, add up all 
the equations, and write B for the arithmetical mean of all the 
S's. The addition gives 

Xn - x^f -\- nB ss n {I ~ a) (2). 



24 PRELIMINARY V 

From which, since Xn and x^ are given, I — a can be calculated. 
soi may then be calculated from the first of the equations (1), 
since ^" has been determined. Having found oo^, x^ is calculated 
from the next equation, and so on successively for all the 
corrections. 



A tube is given to you divided into millimetres (Fig. 11). It 

n 



Fig. 11. 

is to be calibrated between points 1 6 cms. apart, the errors being 
required for points at a distance of 2 cms. apart. That is to say, 
assuming the first and seventeenth centimetre division to be 
correct, the errors at the third, fifth, &c. division are to be 
found. 

The most difficult part of the operation consists in obtaining 
a mercury thread of the required length. It may most easily 
be accomplished by slipping a short piece of indiarubber tubing 
over one end of the glass tube. The free end of the rubber 
tube is then compressed between the finger and thumb of the 
right hand, so as to close it completely, the end of the glass 
capillary tube dipped into the mercury, and then the rest of 
the rubber tube compressed between the finger and thumb of 
the left hand to expel a little air from it. On releasing the 
left hand, a thread of mercury is drawn up. 

The capillary tube should then be quickly placed nearly 
horizontal with the lower end over the mercury in the bottle, 
and the end of the rubber tube released. By gently tilting the 
tube, the thread may be made to approach the lower end of 
the tube, and a small drop of mercury forced out and cut off 
with the finger-nail or a knife. 

By carrying out this operation several times, the thread 
maybe diminished in length till it occupies between 19 and 21 
of the small scale divisions, i.e. within 1 mm. of 2 centimetres. 

When this is the case, move the thread till its left-hand 



V CAUBRATIOX OF A GRADUATED TUBE 25 

end nearly coincides with the division on the left at which the 
calibration is to commence. 

Place the tube on a strip of mirror glass and read both ends 
of the mercury thread by means of a magnifying lens, avoiding 
parallax by placing the eye so that its image is covered by the 
part read and then slipping the lens into position. The excess 
of the observed length of the thread over 20 scale divisions is 
the 81* of the preceding equations. 

Now move the thread forward 20 small scale divisions so 
that its left-hand end nearly occupies the position of the right- 
hand end in the previous case, and obtain 8^^ as before. 

Continue the operation till the ends have been read in 
eight successive positions, and then take readings as the thread 
is moved backwards. The means of the B's observed at each 
part of the scale going and returning should be used in the 
calculations, which should be carried out and tabulated as 
shewn below. 

As in this exercise we only desire to subdivide the distance 
between two given points of the tube into divisions of equal 
volume, we may take the corrections at the initial and final 
points as zero. Putting Xq = x^ = 0, equation (2) becomes 
/ — a = 5. Hence from (1) 
Xi — 8 — 81^, 
x^--x, + 5 - 8,» = (5 - V) -f- (3 - S,»X 

-r,=:a:2+ 3- 53»=(S- V) -f (s -a,o -H (a - 8,% 



= 0. 

In this case x^ is zero, and this will furnish a check on the 
arithmetical calculations. 

A check for some of the intermediate points may be 
obtained by taking threads 4 and 8 cms. long. Calling the 
first division 0, the 4 cms. thread will give the corrections at 
the points 4, 8 and 12, while the 8 cms. thread will only give the 
correction at the division 8. 

To calculate the corrections it is necessary first to find 3, 
which is the mean of the quantities 8i", 8^, &c. If these are 



26 



PRELIMINARY 



tabulated as in the annexed table and the mean found, the 
differences 8 — Sj**, 5 — §2^ may be written down as in the fifth 
column and the corrections obtained by successive addition. 

Arrange your observations and calculations as follows, 
giving the mark on the label attached to the tube so that it 
may be identified : — 

Tube B. Date: 3 Jan. 1898. 



Going 


Returning 


Mean 


&c. 


6-61 
&c. 


Corrections 


- 04 to 2-03 = 2-07 
1-98 to 4-04 = 2-06 
2-06 
2-06 
&c. 2-03 
2-02 
2-01 
2-01 


--05 to 2-0 = 2-08 
&c. 


2-075 
2-065 
2-062 
2-067 
2-032 
2^017 
2-017 
2-005 


•075 
•065 
•062 
•067 
•032 
•017 
•017 
•006 


- -0325 

- -0225 

- -011)5 

- -0245 
+ •OlOo 
+ ^0255 
+ ^0255 
+ ^0375 


- •033 = ^2 

- •055 = ^4 

- •076 = a:g 

- -099 = 0:8 
-•089 = a:i, 
-•063=a;i3 

- •038 = a;i4 
-•000 = a;ie 




Sum = 


•340 








Mean = 5= 


•0425 




-•05to4-10 = 4-15 

4-16 

&c. 4-09 

4-03 


--06 to 4-10 = 4-16 
&c. 


4-155 
4-160 
4-095 
4-005 


•155 
•160 
•095 
•005 


-•041 
-•056 
+ •009 
+ •099 


-•051 = a;4 
-•107 = ^8 

- •098 = :ri2 
-•00 =x,. 




Sum = 


•415 








Mean = 5 = 


-104 




-•03 to 8-12 = 8-15 
7-97 


-•04 to 8-12=8-16 


8-155 
7^955 


•155 
-045 


-•100 
+ •100 


- •10 = ^8 




Sum = 


•110 








Mean=5= 


•055 





Taking the readings along the tube as abscissae and the 
corrections at these readings as found by the first measurements 
as ordinates, above or below according as the correction is 
positive or negative, plot a " Calibration Curve " for the tube 
as shewn below, Fig. 12. 

Do the same for the second and third sets of readings, 
taking the same abscissae. 



CALIBRATION OF A GRADUATED TUBE 



27 



If I is the length of the mercury thread in any position, and 
A the mean area of the cross 
section of the tube within the 
part occupied by the mercury 
thread, Al is the volume of the 
thread. As this volume remains 
constant as the thread is moved, 
the length of the thread varies 
inversely as the mean area of 
cross section within its length. 

Hence by taking again dis- 
tances along the tube as abscis- 
sae, and erecting at the different points occupied by the centre 
of the mercury thread ordinates inversely proportional to the 
observed lengths of the threads, we get a representation of the 
way in which the cross section of the tube varies. 











V^ ' 1 ' J 


-Sx i 


S> J ■' 


- 5^ ^ H- 


:_^t. _s - - -_ -t:,--- 


^ I • - 


\ /-s- 




^S t J. 


M.. 5^-2. _L :_ _ 


HB--- ^^-__L 


: t::: 


±_ 



Fig. 12. 



Centre of 


Length of 


Reciprocal of 


thread 


thread 


length 


1 cms. 


2075 cms. 


•482 


3 


•065 


•484 


5 


•062 


•485 


7 


•067 


•484 


9 


•032 


•492 


11 


•017 


•495 


13 


•017 


•496 


15 


2-005 


•499 



Plot these reciprocals as shewn below, Fig. 13. 

The most important practical application of the preceding 
exercise occurs in the calibra- 
tion of thermometer tubes. All 
thermometers used for accurate 
work should be calibrated either 
by the maker or by the observer. 
Information as to the methods 
of breaking off a mercury thread 
and complete methods of cali- 
bration will be found in Guil- 

laume's Thermoniiirie, 

Fig. 18. 



|..«4»- ;:t-- 

pi: 

\ :^ii 

_L -,: 

_^4^-^ 4 

^tX.— 'Ziml. 

ZIhh---^-----**- HI--- 



BOOK II. 

MECHANICS AND GENERAL PHYSICS. 




Fig. 14. 



SECTION VI. 

THE BALANCE. 

Apparatus required: Delicate balance, centigram rider, 
two 500 gram zveights. 

The Balance, in its simplest form, consists of a straight beam 
AB (Fig. 14) provided at its 
centre with a knife edge G 
on which it is supported, and 
carrying at its ends the pans 
P and Q, on which the masses 
to be compared are placed. 
If the two halves of the beam 
are alike in every respect, so that the centre of gi^avity of the 
beam is at the point of support G, and if the pans have equal 
mass, the balance will be in neutral equilibrium, whether 
unloaded or loaded with equal masses. If unequal masses are 
placed in the pans, the equilibrium will be unstable, however 
small the inequality may be. 

It would be inconvenient to have a delicate balance con- 
structed according to this principle, for as two masses are never 
exactly equal, the balance would never be in equilibrium. 
A delicate balance, to be useful, should allow us to determine 
the difference between two 
nearly equal masses, and it is 
proved in treatises on Me- 
chanics that this can be done 
by constructing the beam AB 
of the balance so that its centre Fig. 15. 




VI 



THE BALANCE 



29 



of gravity is slightly below the fulcrum G. The balance will 
then have a stable position of rest even if the two weights 
P and Q are not quite equal, and the amount of inequality 
may be determined by an observation of this position. 

The pans of a delicate balance are suspended from agate 
planes supported on knife edges at the ends of the beam, which 
is itself provided at its centre with a knife edge resting on 
an agate plane at the top of the pillar (Fig. 15). The accuracy 
which can be attained in weighing, depends to a great extent 
on the freedom with which these three knife edges can turn on 
their planes. In order to save these delicate parts as much as 
possible from wear and tear, a so-called arrestment is provided, 
by means of which the central knife edge of the beam may be 
raised from its agate plane, while at the same time the agate 
planes of the pans are raised from their knife edges. The 
balance is enclosed in a glass case, and the aiTestment worked 
from the outside by the screw head in front of the case. 
Wlienever the balance is not in use the arrestment should be 
raised. The change of position of the beam of the balance is 
observed by the help of a pointer attached to the beam, the free 
end of which moves in front of a scale at the foot of the pillar. 

Before beginning to work, students should carefully inspect 




T 



n 



Fig. 16. 



^ 



30 MECHANICS AND GENERAL PHYSICS VI 

the balance (without opening the case), and make in their note- 
book a rough sketch of the mechanism of suspension and 
arrestment. The reference books in the laboratory library 
contain detailed descriptions of the balance, which may be 
consulted by the students with advantage. 

Exercise I. To find the zero of the unloaded 
balance. 

Carefully lower the arrestment of the balance, and watch 
the movement of the pointer. If it moves slowly through a 
range of from 3 to 6 divisions of the scale, observations may be 
commenced. If the extent of the swing is greater, it is a sign 
that the arrestment was not handled with sufficient care ; it 
should be gently raised again, and lowered more slowly. If the 
swiog is too small, open one of the windows of the balance and 
produce a weak current of air by gently waving your hand 
a few times inside the case, being careful not to bring it into 
contact with any part of the balance. A little practice will 
enable you to obtain a workable swing. The window is then 
shut and the observations commenced. 

As it would take too long to wait until the balance has 
come to rest, the position of rest must be determined by 
observations made during its motion. 

For this purpose the turning points of the pointer on the 
scale must be observed. If the balance moved without any 
friction, the pointer would move to equal distances on both 
sides of the position of rest, which could therefore easily be 
deduced from two successive readings of the turning points. 
Owing to friction the oscillations gradually diminish, and in 
order to find the position of rest more than two readings are 
required. 

The following example will shew how the position of rest 
of the balance is calculated from 

three observations of the turning i i I *^ I i I 

points of the pointer. I ■ ■ I I I M I I I I I I I I I I I I I 

o ^i. ^r u ^ri. 5 10 15 20 

Suppose that after the ar- 
restment has been carefully ^^' 



VI THE BALANCE 31 

lowered, the pointer moves to the right, then turns back again 
to the left at a point which is not observed. Let the three 
following turning points, indicated by arrows in Fig. 17, be 
read off as follows: ll'O, 168, 11*7, the centre of the scale 
over which the pointer moves being marked 10 in order to 
avoid negative numbers. If there were no friction the arith- 
metical mean between 110 and 16*8 would give the position of 
equilibrium. In reality this mean (13*9) gives too small a 
value, for without friction the needle would have moved beyond 
16'8. For the same reason the mean of 16'8 and 117 (14*25) 
would give too large a value, and as the frictional retardation 
has been practically the same during the forward and during 
the backward swing, the errors of the two means will be equal 
and in opposite directions, and hence the arithmetical mean 
between 13'9 and 14*25, i.e. 1408, gives a sufficiently correct 
result for the required position of rest. The same numerical 
result is obtained by taking the mean of the two successive 
swings on one side (1185 in the above example) and then the 
mean between this number and the intermediate turning point 
on the other side. Thus the mean of 11 35 and 16 8, or 1408, 
would be the true position of rest. If the numbers in the first 
instance be read off correctly to a tenth of a division, the final 
result involving three readings may be given to another decimal 
place, for though the observations may not be sufficiently 
accurate to fix the final position within a htmdredth of a scale 
division, they should be sufficiently accurate to determine it 
more nearly than a tenth of a division. Practice will enable 
the student to read with certainty to less than a tenth of a 
division. 

The effect of small errors of observation on the final result 
will be diminished by taking more than three readings, care 
being taken, however, to take an uneven number, beginning 
and ending on the same side. The same method of reduction 
may then be employed. There is, however, a considerable loss 
of time when too many readings are taken, the time being 
better employed in taking a fresh observation altogether, and 
unless there are special reasons to the contrary, students in the 
laboratory should take only five successive readings and put 



32 



MECHANICS AND GENERAL PHYSICS 



VI 



them down in their note-book as in the following example, 
noting the temperature indicated by the thermometer in the 
balance case. 



18 June 1896. Balance: A. 





Turning points 




left 


right 




7-4 

7-6 

7-7 


131 
130 


Sums 
Means 


22-7 
7-57 


26-1 
13-05 


Position of 
rest 


10-31 



Temperature 18°-1 C. 



Whenever a weighing is taken the observations should be 
recorded as above, but in writing out results, the final positions 
of rest and the temperatures need only be given. 

To bring the balance to rest raise the arrestment gently 
so that it lifts the beam when the pointer is passing through 
the point of rest, and thus injures the knife edge as little as 
possible. 

Take thi-ee sets of observations of the position of rest of the 
balance, raising the arrestment between each set. Find the mean 
value of the zero as determined by the three sets. 

The temperature of the balance case is required in this and 
the next exercise, in order that it may be ascertained by 
observations taken on different days whether the zero of the 
balance shews any appreciable change depending on temperature. 
For ordinary purposes it is not usual or necessary to observe 
the temperature. 



VI THE BALANCE 33 



Exercise II. To find the sensibility of the unloaded 

balance. 

The change produced in the position of rest of an unloaded 
balance by 1 milligram excess of weight on one side, is 
known as the "sensibility" of the unloaded balance. The 
sensibility varies to some extent with the load placed on 
the pans. In order to produce a small excess of weight (less 
than 1 centigram) a rider is provided, which, by means of a 
sliding rod, may be shifted along the beam. The weight of the 
rider is 1 centigram ^ but the nearer it is to the fulcrum of 
the balance, the smaller will be its turning moment ; each half 
of the beam is divided into 10 equal parts marked 1, 2... from 
the centre to the end, so that if the rider stands at say 2, its 
effect is the same as the addition of 2 milligi-ams to the weights 
on the pan. 

To find the sensibility it will in general be suflScient, by 
means of the rider, to add to one side a weight equivalent to 
1 mgrm. (or for sake of greater accuracy 2 mgi-ms. dividing the 
resulting change of position of the pointer by two). In order 
to obtain suflScient practice in accurate weighing, the student 
is required to determine the positions of rest, varying the excess- 
of weight from 8 mgrms. on one side, to 8 mgrms. on the other 
of the centre of the beam. If, however, the balance is very 
sensitive, it may not be possible to keep the swings within the 
limit prescribed above (6 scale divisions), when an excess of 
6 or 8 mgrms. is placed on one side. 

The observations should be taken and recorded as in 
Exercise I. The numbere obtained should bt; siinimarised in 
the Book of Results as follows : — 



» The weights of riders sold commercially often differ so much fh)m their 
nominal value that serious errors may be introduced if they are used without 
being tested. 



8. P. 



34 



MECHANICS AND GENERAL PHYSICS 

20 June 1896. Balance: A. 



VI 



Time, 10 h. 10 m. 






Temperature of Balance Case, 


l8°-2 C. 


Sensibility 


Mgs. excess of Weight 


Position of Rest 


Difference 





10-07 






On left side : 8 


12-79 






6 


12-17 


•62 




4 


11-43 


•74 




2 


10-81 


•62 





On right side : 2 


10-13 
9-46 


•681 
•67/ 


•338 


4 


8-81 


•65 




6 


8-11 


•70 




8 


7-42 


•69 







10-15 




1 


Temperature of Balance Case, 


L8°-9C. 


Time, 10 h. 50 m. 







Position of rest at beginning of experiment ... 1007 

„ „ end „ 1015 

Temperature at beginning „ 18^-2 C. 

end „ 18°-9 

Sensibility of balance '338 

Note. — In calculating the sensibility of the balance , the only 
numbers in the above table taken into account, are those which 
give the deflections produced by 2 mgrms. on either side of the 
zero of the balance. 

The sensibility of the balance may be increased or dimi- 
nished by raising or lowering the centre of gravity of the beam, 
and each balance is provided with an adjustment for this 
purpose. But it must be remembered that an increase in the 
sensibility does not necessarily mean an increase in the accuracy 
with which a weighing can be made. As the balance is made 
more sensitive, its time of vibration increases, each weighing 
consequently takes a longer time, and small changes in the zero 
of the balance may take place owing to changes of temperature 
or other causes. The longer the time, the greater the proba- 



VI THE BALANCE 35 

bility of the occurrence of such disturbances. Moreover, the 
extra time thus spent might just as well be spent in repeating 
the weighing with the former sensibility, and the result would 
probably be improved more in this way than in the other. 
There is always a limit to the accuracy with which a balance 
will weigh, and once that degree of accuracy has been attained 
by a careful reading of the position of rest, it does more harm 
than good to attempt to increase the sensibility. Students 
will obtain the best results by carefully practising the method 
of obtaining the position of rest which has been explained 
above, depending on accuracy of reading rather than on a great 
sensibility of their balance. 

Exercise III. To find the zero of the loaded balance. 

If two exactly equal masses were available, to tind the 
zero of the balance when loaded with them, it would simply 
be necessary to find the position of rest when the masses 
were placed in the pans. As however absolute equality is not 
easily attainable, we must find a way to determine the zero 
notwithstanding the inequality of the masses. If the position 
of rest has been found with two weights, P and Q, in the pans, 
then if the weights were equal, on interchanging them the 
position of rest would be exactly the same, and this would be 
the case independently of any adjustment of the balance. Even 
if the arms are not equally long, or if they alter their length by 
bending, two equal weights may still be interchanged without 
change in the position of rest. 

But suppose the position of rest is 12-3 when P is on the 
left-hand side, and 107 when P is on the right. If Q were 
slightly incrccosed, the first number would be diminished and 
the second number increased by equal amounts, and the 
difference in the two readings would diminish. It follows that 
if the increase in Q were such that the new position of rest is 
the arithmetical mean between 12*3 and 10*7, i.e. 11 "5, no 
change would be produced on interchanging the weights, and 
hence 11*5 is the position which the balance would take up if 
equal weights P were placed in the pans. 

If the sensibility of the balance were known, the experiment 

3—2 



86 



MECHANICS AND GENERAL PHYSICS 



VI 



would give the difference between P and Q, which is evidently 
the weight required to be added to Q in order to produce a 
difference of 12*3 — 11*5 = "8 scale division. 

Two masses weighing nearly 500 grams are provided. Com- 
pare the position of rest when no weights are in the pans, with 
the position of rest when these equal masses are placed in them. 

Observe the temperature. 

Take one observation without weights, then one with the 
weight marked 1 in the left and that marked 2 in the right 
pan, one with the weights reversed, then one with the weights 
as at first, then one without weights. 

Again observe the temperature. 

Note. — If the masses are not placed quite centrally on the 
scale pans, the pans will oscillate, and errors may thus be 
introduced. In accurate weighing this should be avoided as 
much as possible, by arranging the weights symmetrically. 
Oscillations may be stopped before the arrestment is lowered, 
by carefully placing the open hand so as just to touch the pan 
with thumb and forefinger, or by touching it with a camel hair 
brush. 

The results are entered as follows : 

20 June 1896. Balance: A. 



Temperature 


.. 18°-5C 


Zero of balance without load 


.. 11-34 


Weight no. 1 on left pan ; position of rest . 


.. 1010 


a ^ i> )) 


.. 11-42 


i) ^ }} }) 


.. 1014 


Zero of balance without load 


.. 11-44 


Temperature 


.. 18°-9C. 


Mean temperature 


.. 18°-7C. 


Zero of balance for load 500 grs 


.. 10-77^ 


Mean zero of balance without load ... . 


.. 11-39 


Change of zero 


•62 


Change of temperature 


•4°C. 



1 This number is obtained by combining the mean of the readings found 
with weight no. 1 on the left pan, with the reading found when that weight was 
on the right pan. 




VI THE BALANCE 37 

Exercise IV. To find the sensibility of the loaded 
balance. 

In Fig. 15 the three knife edges are represented as being 
in the same straight line, but 
this is not necessarily the case. 
Let A and J5, Fig. 18, represent 
the knife edges from which the 
pans are suspended, and C the 
knife edge on which the beam 
of the balance rests. We shall 
assume that by means of the ^' 

movable vane attached to the beam, the centre of gravit}^ G, 
of the beam alone, has been adjusted so that when it is 
vertically beneath C the line AB i^ horizontal. Let the dis- 
tances of G and of the line AB below C be h and k respectively. 
Then if P be the weight of the pan and contents, and a the 
length of the arm on the left, and Q that of pan and contents, 
and h the arm on the right, and if 6 be the rotation of the beam 
produced, we have, taking moments about C, 

P (a cos e-k sin 0)^Q{b cos d + k sin 6) + Wh sin 6, 

, ^_ Pa-Qb 

If Q = P -\-j), then since as a rule b will be equal to a, and k 
will be small, the expression may be written, neglecting pk in 
the denominator : — 

The deflection 6 produced by a given excess of weight p on 
one side will therefore diminish with the load if k is positive, 
and increase with the load if k is negative. But the value of k 
itself will vary with the load, as the beam of the balance always 
bends a little, and the knife edges descend as the load is 
increased. The makers sometimes adjust the knife edges so 
that the line AB is slightly above C when the balance is 
unloaded, and below C when the balance is loaded with more 
than half the maximum load for which it is intended. In that 



38 MECHANICS AXD GENERAL PHYSICS VI 

case the sensibility will first increase and will then decrease 
with the load. 

Theoretical Exercise for Advanced Students. — Writing in- 
stead of k, — k^-h A.P in order to indicate its change with the 
load, prove that the maximum sensitiveness will be reached 

k 
for the weight P' = ~ ; and that A^o and \ are determined by 

the equations 

2\ (Pi - P,) (Pi + P. - 2P') = pa (cot e, - cot e,\ 
ko (Pi - P2) (Pi + P2 - 2P0 = P'pa (cot (9i - cot 0^), 
where a represents the length of the beam, and p the additional 
weight which produces deflections 6^ and 6^ when the loads are 
Pi and P2 respectively. 

With the help of these equations the quantities ko and \ 
could be calculated ; but the result depends on the assumption 
that the bending of the beam is symmetrical and proportional 
to the load, and that it does not throw the centre of gravity of 
the beam to one side. 

Determine the positions of rest of the balance under the 
following conditions : — 

(1) The two 500 gram weights in the pans. 

(2) An excess of 2 mgrs. on the left. 

(3) Asin(l). 

(4) An excess of 2 mgrs. on the right. 

(5) As in (1). 

Enter your results as follows : — 

7 Oct 1896. Balance: A. 
Time at beginning of experiment, 11 h. 15 m. 
Position of rest 

(1) with load of 500 grms 7-98] 

(2) „ + excess of 2 mgrs. on left 8-381 8-04 '34 

(3) „ ... 8I0J] 

(4) „ + excess of 2 mrgs. on right 7'84 >8*19 '35 

(5) „ 8-28 J 

Time at end of experiment, 11 h. 32 m. 

Mean sensibility '172. 



VI THE BALANCE 39 

Exercise V. To determine the ratio of the arms 
of the balance. 

If we assume that the ratio of the lengths of the arms of 

a balance is not affected by the bending of the beams, the 

change of zero with the load can only be due to an inequality in 

the arms of the balance, and the data obtained in Exercises III. 

and IV. are sufficient to determine that ratio. In Exercise III. 

the change of zero from no load to one of 500 grams was found 

to be '62 divisions, and in Exercise IV. the sensibility was 

found to be '17 ; hence to bring the loaded balance with its 

zero 10*77 to the unloaded zero 1139 we should have to add 

'62 

-T- = 3*6 mgrs. on the left-hand side. Let this quantity be 

denoted by p, and let the lengths of the left and right arms 
of the balance be a and b respectively. When the loaded 
balance comes to rest, the moments which act on the beam will 
be the same as in the case of the unloaded balance, with the 
addition of P •¥ pa on the left side, and Pb on the right side, 
and if the position of equilibrium is unaltered it follows that 

(P + p)a = Pb, 
or 

a P oOO 

As the beam of the balance has a length of about 22 cms., 
the error of adjustment of the knife edges only amounts to 
•00016 cms. 



SECTION VII. 

ACCURATE WEIGHING WITH THE BALANCE. 

Apparatus required : Balance, piece of quartz, box of 
weights. 

The method to be adopted in weighing depends on the 
object for which it is carried out. Extreme accuracy always 
means the spending of a good deal of time on the observations, 
which would be wasted if from the nature of the case such 
accuracy were unnecessary. 

In chemical analysis relative weights only are required, and 
an inequality of the arms of the balance will not affect these so 
long as the weighings are carried out on the same side, and the 
method of Exercise I. p. 30 may therefore be adopted. More- 
over, in many cases the errors introduced by impurities in the 
substances or by other causes, may amount to several mgrms., 
and as it would then be absurd to conduct the weighing cor- 
rectly to the 10th part of a mgi-m., the method may be further 
shortened, by reading three turning points instead of five, by 
assuming the zero to remain constant during the series of 
weighings so that it need only be taken once, and especially 
by using a less sensitive balance having a shorter time of 
vibration. 

If the same balance is always used, much time may be 
saved by determining its sensitiveness for different loads once 
for all. Unless the adjustment is altered, the sensitiveness will 
remain the same for a considerable period. 

The custom of assuming the zero always to be at the centre 
of the scale, is not to be commended, as it may cause serious 
errors. The student should first learn to weigh accurately 



VII ACCURATE WEIGHING WITH THE BALANCE 41 

irrespective of the time it takes; he will gradually learn to 
weigh quickly, and to know how to save time when great 
accuracy is not required. In all cases where absolute and not 
merely relative weights are required, some method should be 
used which eliminates all errors of the balance, and corrections 
must also be made for the upward pressure of the air in which 
the weighing is conducted. We proceed to explain the way in 
which this is done. 

Corrections to he applied to weighings for the buoyancy 
of the air. 

When a body is surrounded by air, it is acted on by an 
upward force equal to the weight of the air it displaces. Two 
bodies having equal masses but different densities, will occupy 
different volumes, and if these bodies are placed on the two 
pans of the balance, it will show an apparent inequality in the 
weights owing to the difference in the upward pressure of the 
air on the two bodies. On the other hand, if two masses of 
different densities balance each other completely when placed 
on the pans, they will not in reality be equal. 

If M is the mass of the substance weighed, and p its density, 
its volume is Mjp, and the upward force of the air will be equal 
to the weight of a mass MXjp of air, where X is the density of 
the air. 

Hence the resultant downward force which acts on the 
beam of the balance, is the same as if the surrounding air had 
been removed and a mass M{\ —X/p) placed in the pan. Simi- 
larly a mass W of density a- in the second pan, produces the 
same downward force as a mass W{1 — X/<r) suspended in vacuo. 
If the balance, supjxjsed to possess equal arms, is in equilibrium, 
we have : — 

il/(l-X/p)=F(l-X/o-). 
Hence 

if = F (1 - X/<r)/(l - X/p) = Tr(l - \/cr + X/p) approx. 

The last approximate result is obtained by neglecting the 
squares of the small quantities X/p and X/a, which is in this 
case allowable (see Intermediate Practical Physics, pp. 14 
and 15). Hence the quantity which has to be added to the 



42 MECHANICS AND GENERAL PHYSICS VII 

apparent mass W of the weights to obtain the true mass M of 
the body weighed is 

when (T represents the density of the weights, generally brass, 
and may be taken to be 8"4. The density \ will vary with 
the pressure and temperature of the air, and the amount of 
moisture present. It will be seen that the correction is positive 
or negative, according as the density of the substance weighed 
is less than or greater than that of brass. If the masses on one 
or both sides of the balance consist of different materials, the 
correction must be determined separately for each. Thus if 
the weights are partly of brass and partly of platinum, it may 
be necessary to take this into account, and the correction be- 
comes 






when Wi, o-^ are the mass and density of the brass weights, and 
TFs) o"2 those of the platinum weights. 

In the example given to illustrate Exercise II. of the present 
section, a mass of quartz weighing nearly 200 grms. is weighed 
to a tenth of a milligram, i.e., to about one part in two millions. 
The density of quartz being 2'65, the correction for buoyancy 
amounts to about 68 mgrms., and has therefore to be determined 
with an accuracy of one part in 680. This means that the 
temperature of the air in which the quartz is weighed, must be 
known to J of a degree, and its pressure to 1 millimetre of 
mercury. The amount of moisture present in the air should 
also be known, but no appreciable error will be committed if 
the air is assumed to be half saturated. It may also be verified 
that an error of one part in a thousand in the assumed densities 
of the quartz and the brass weights, would cause errors of '1 and 
'03 mgrm. respectively in the weighing. This shews how very 
difficult it is to weigh accurately to one part in a million. 

Before passing on to the Exercises in weighing, students 
should read carefully through the following instructions, which 
must be rigidly adhered to, as otherwise the balance may be 
seriously damaged. 



VII ACCURATE WEIGHING WITH THE BALANCE 43 

Precautions necessary in weighing with a delicate balance. 

1. Test whether the balance is in working condition, by 
lowering the arrestment carefully ; the pointer should slowly 
swing through a few divisions only. 

2. See that the box of weights is complete, then place the 
riders on their supporting arms. 

3. Do not touch the weights with your fingers but with the 
pincers or forks provided. 

4. Xever place weights on the pans or take weights ojf, except 
when the balance is arrested. 

5. The arrestment must be lowered with special care during 
the first stages of weighing, when the weights on the two sides 
are not yet nearly equal. Watch the pointer while the arrest- 
ment is lowered very slowly. As soon as the pointer is seen to 
start sharply to one side, raise the arrestment. Notice carefully 
whether the motion of the pointer to the left or right means 
that the weights placed in the pan are too small or too great. 

6. If the arrestment is to be raised while the balance is 
swinging, wait till the pointer is nearly at its central position, 
then raise. The least possible injury will in this way be done 
to the knife edges. 

7. The final weighings must be made with the balance case 
closed, and Ccire must be taken that the pans do not swing. 
Large swings of the pans should be carefully stopped by touching 
the pans with thumb and forefinger or with a camel hair brush 
while the beam is arrested. 

8. In reading successive turning points take no account of 
the first, which is sometimes irregular. 

9. When the weighing is complete, replace the weights 
carefully into their proper places in the box, remove the 
arrestment handle, place it in the balance case, and close the 
case. 

Exercise I. To weigh a body using the zero at 
no load. 

1. Find the position of rest of the balance without load. 
Call this the zero of the balance at no load for the time being. 



44 MECHANICS AND GENERAL PHYSICS VII 

2. First ascertain by trial on a rough balance, that the weight 
of the given quartz crystal is say between 100 and 200 grms., 
then place it in one pan (the left for instance) and weigh to the 
nearest centigram in the following way : — Put 100 grms. in the 
right pan and add the weight which comes next in descending 
order of magnitude in the box of weights. Continue adding 
weights as you have been taught to do in the Intermediate 
Course (p. 50 and 51), until you find that the addition of 
another centigram shews excess of weight. Determine the 
weights on the pan, by noting the empty compartments in the 
box of weights, and record in your note-book. The number 
found must be checked by noting the weights themselves as 
they are removed from the pan at the end of the experiment. 

3. Determine the weight to the nearest 2 milligrams. 
Use the rider for this purpose, placing it first at the point 

marked 6 on the beam, then at 8 or at 4, according as the 
additional 6 mgrms. have been found too small or too great. 
The student will have been able to proceed so far without 
taking readings of the pointer. But at this stage he will have 
to make a rough determination of the position of rest by taking 
the arithmetical mean of the readings of two consecutive turning 
points. As he gets nearer to the true value of the weight, he 
will have to determine the position of rest more accurately, and 
it will be necessary to read five turning points. 

Let it be found in this way, that the weight lies between 
185-874 and 185-876 grms. 

4. Determine the weight to the tenth part of a milligram. 
Observe accurately the positions of rest for the two weights 

differing by 2 milligrams between which the true value has 
just been found to lie. Then by interpolation calculate the 
weight which would bring the position of rest to the zero at no 
load. With very delicate balances it may be necessary to deter- 
mine the weight to the nearest milligram before proceeding to 
interpolation. 

5. Again determine the zero at no load. 
Enter vour results as follows : — 



VII ACCURATE WEIGHING WITH THE BALANCE 45 



5 Oct 1896. Balance: A. 

Zero of unloaded balance 

Position of rest with 185-878 grms. in right pan 
„ „ ,, » + 2 mgrms. „ „ 



10-36 
11-58 

9-89 
10-34 
10-35 

1-69 



Zero of unloaded balance 

Mean zero at no load ... 

Difference produced by 2 mgrms. 11-58-9-89 = 

Additional weight required in scale divisions 1 1-58 - 10-35 = 1-23 

2x1-23 
„ m mgrms. ^ ^^ = ... I'O mgrs. 

Required weight ... ... ... ... 185-8795 grams. 

Exercise II. To weigh a body by the method of 
interchanges, sometimes called Gauss's method. 

The method given in Exercise I. does not correct for any of 
the errors of the balance, and if the weight is to be obtained 
accurately, it is necessary to adopt the method given in the 
present exercise, or one equivalent to it. The method, which 
has already been used in Exercise III. of the previous section, 
consists in interchanging the weights in the pans, and finding 
directly, or by interpolation, a weight which will bring the 
balance to the same position of rest, whether the substance to 
be weighed is in the right or in the left pan. It is clear that 
two weights which can be interchanged without altering the 
position of rest of the balance, must be equal. 

Proceed as follows : — 

1. Find the weight to the neare.st 2 mgrms., as in Exer- 
cise I. 

2. Find the position of rest when the substance is placed 
in the left pan, the lower one of the two limiting weights being 
placed on the right-hand side of the beam. 

3. Interchange weights and substance and find the position 
of rest, being careful to remove the rider from the right-hand 
beam, and to place it or a .similar one at the corresponding point 
on the left-hand beam. 

4. Interchange once more, so as to bring back the substance 
into the pan in which it was originally placed. 

5. Increase the weight on the right side by 2 mgrms., 
and determine the position of rest. 



46 MECHANICS AND GENERAL PHYSICS VII 

Reduce and enter as follows : 

5 Oct. 1896. Balance: A. 

Position of rest with 185-878 grm. weights on right ... 10*57 (a) 

left ... 1000(6) 

„ „ right ... 10-45 (c) 

„ „ „ +2 mgrms. „ „ ... 8-94 

Difference produced by 2 mgrms. ... ... .. 1-51 

Mean of (a) and (6) 10-51 

Zero of balance with load ... ... ... 10-26 

Ji 

Additional weight required in scale divisions 10-26 - 1000 -26 

2 X -26 _ 

„ „ „ „ mgrms. -p^ = ...-34 mgrms. 

Required weight ... ... ... ... ... 185-87834 grams. 

Note. — It may happen, if the arms of the balance are not 
sufficiently equal, that the weight does not in reality lie between 
the two limits found in Exercise I. If the real weight is above 
the higher limit, the additional weight required to produce the 
balance will be found greater than 2 mgrms. If, on the other 
hand, the true weight is smaller than the lower limit, the posi- 
tion of rest when the weights are on the right ((a) and (c) 
above) gives a lower reading than in (h) when the weights are 
in the left pan. Thus, suppose the reading for (6) had been 
H'O instead of lO'O we should have had to write : 
Additional weight required in scale divisions 10-26 — 11*00=— "74, 
the negative sign indicating that the correction has to be sub- 
tracted from 185*878. Students should note carefully whether 
the correction they find is to be added or subtracted. 

It will be observed that the substance is weighed twice 
in one pan and once in the other. The object of this is to 
eliminate the effects of changes in the balance which take place 
if the temperature of the balance case is increasing owing to 
the approach of the observer, or to the presence of gas flames. 
The difference between (a) and (c) in the above example, may 
either be due to accidental causes or to a systematic change. 
If the former, the mean will be a more probable value of the 
position of rest than either of the observed numbers ; if the 
latter, the mean will represent the position of rest at the time 



VII ACCURATE WEIGHING WITH THE BALANCE 47 

at which the observation (6) was taken, provided the observa- 
tions were carried out at nearly equal intervals of time. In any 
case to get the best result the mean of the first and third 
observations should be compared with the second. If extreme 
accuracy is not required, the third observation may be dis- 
pensed with ; if, on the other hand, it is required to determine 
the position of rest correctly to the hundredth part of a scale 
division, it will be necessary to interchange the weights 
oftener. 

To complete the exercise, the result should be checked by 
placing the rider at the position corresponding to the weight 
found to the nearest tenth of a milligram (185*8783), and the 
weighing repeated. The barometer and thermometer should 
also be read, so that the buoyancy correction may be applied. 
The barometer need only be read once, unless there is reason to 
suppose that it is rapidly changing at the time, but the tem- 
perature of the balance case should be taken at the beginning 
and end of the experiment. An example will shew how the 
final result is now arrived at. 

5 Oct., 1896. Balance: A. 

Assumed weight 185-8783 grs. 

Barometer 7 6 '4 cm. 

Time 11 h. 15 m. Temperature of balance case .. ... 18° -6 0. 

Position of rest with 185-8783 grs. on right 10-36 

left 10-46 

right 10-39 

+ 2 mgrms. „ 8-91 

grs. , 10-41 

11 h. 32 m. Temperature of balance case ... ... 18''-8 C. 

Mean position with weights on right (10*36 + 10-39)/2 = 10-375 

left 10-46 

Zero of loaded balance = (10375 + 10-46)/2 ... = 10-42 

Difference produced by 2 mgrms. = 1040 - 8 91 = 1-49 

Additional weight required in scale divisionsl 

10-42-10-46 J ••• 

2 X *04 

Additional weight required in mgrms. . . - ... = '05 mgrms. 

Required weight 185*8783 - 00005 = 185*8783 grams. 



» 


>» 


»> 


» 


)) 


»> 


>> 


>> 



48 



MECHANICS AND GENERAL PHYSICS 



VII 



It will be seen that the repetition of the experiment has led 
to a result which is the same as that of the previous determina- 
tion, and if the experiments have been carried out with care, the 
difference between them should never exceed '2 mgrm. If the 
difference exceeds 25 mgrm. the rider should be placed at the 
position indicated by the last result, and a fresh determina- 
tion made. 

Buoyancy Correction. 

To calculate the buoyancy correction, first calculate by the 
following method the density of the air, assuming it to be half 
saturated. 

TABLE I. 

Density of dry air at ordinary temperatures and pressures. 



Temperature 


Pressure in cms. of mercury 


73 cm. 


74 cm. 


75 cm. 


76 cm. 


77 cm. 


10° c. 
15 
20 
25 


•001198 
1177 
1157 
1138 


1214 
1193 
1173 
1153 


1231 
1209 
1189 
1169 


1247 
1226 
1205 
1184 


1264 
1242 
1220 
1200 



TABLE IL 

Maximum pressure of water vapour 
at ordinary temperatures. 



Temperature 


Pressure =p 


10° C. 


•91 cms. Hg. 


12 


1-04 „ 


14 


M9 „ 


16 


1-35 „ 


18 


1-53 „ 


20 


1-74 


22 


1-96 


24 


2-22 „ 


26 


2-50 



VII ACCURATE WEIGHING WITH THE BALANCE 49 

Consider a mixture of two gases the densities of which at 
a pressure P and given temperature, are d^ and rf^. Let the 
partial pressures of the two gases be p, and p^ respectively, 
the total pressure being P=^pi-hpi. The density X of the 
mixture is 

i.e., the density of the mixture is equal to that which the 
first constituent alone would have at the pressure 

If the two gases are air and aqueous vapour respectively, the 
index 2 referring to the latter, the ratio d^/d^ is very nearly 5/8, 

hence F + —j-^ p^^ P - ^p^. 

If then p^, the pressure of the aqueous vapour present in the 
atmosphere at the time, is known, we may use Table I. to 
determine the density X, by taking the air to be dry at a pressure 

3 
P— -pi, instead of saturated at the observed barometric pres- 
sure P, and if we assume the air to be half saturated, i.e., take 
P2 = ^, p being the maximum pressure possible at the observed 

temperature, given in Table II., we should have to take the 

3 

equivalent air pressure to be P — j^p. The value of p at the 

temperature of the room will on the average be about 15 mm.; 
the error we should make if the air happened to be totally 
dry or totally moist, would therefore be the same as if we had 
measured the height of the barometer incorrectly by about 
3 mm., which would cause an error in the density of the air and 
in the buoyancy correction, of about one part in 250. If we 
require to make the correction with certainty to less than that 
amount, we should have to measure the pressure of aqueous 
vapour in the balance case. 

In the above example the barometer stood at 7638 cm., and 
the mean temperature of the balance case was 18°'7C. 

8. p. 4 



50 MECHANICS AND GENERAL PHYSICS VII 

Pressure of aqueous vapour at 18°7 C. ... .,, = 1'6 cm. 



(n)" 



= -3 



P =76-38,, 

3 

P — Y^j9 (to the nearest millimetre) ... ... =76'1 „ 

Density of dry air at 18° C. and 76*1 cms. pressure 

(by interpolation) from Table I. ... ... ='001215 

Density of dry air at 19" C. and 76*1 cms. pressure 

(by interpolation) from Table I. ... ... ="001211 

Density of dry air at 18°*7 C. and 76*1 cms. pressure 

(by interpolation) from last two values ... ='001212 



To find the buoyancy correction X ^ 



the 



specific gravity of the body weighed must be known approxi- 
mately. In the above example the body was Quartz, so that 
if W and p refer to the body weighed, Tfi and a^ to the brass 
weights, and TTg, a^ to the platinum weights, 

Z =185:88 

p 2-653 

Ii = l^ = 22-021 

<"' 8'*^ [ = 22-06 

I?=:!i = -04 

<r, 21-5 ) 



Z_Z._I? =48-00 

.-.buoyancy correction = '001212 x 48'00 = '05817 
Weight found = 185'87825 

Weight corrected (to the nearest tenth 

ofamgrm.) =185*9364 grams. 

It will be seen that if we had neglected the platinum weights, 
we should have made an error of about 1/20 mgrm., which, con- 
sidering the uncertainty in the assumed specific gravity of the 



VII ACCURATE WEIGHING WITH THE BALANCE 51 

brass weights used, would have been allowable, but if we had 
assumed all the weights 185*88 to be brass, the error would 
have amounted to more than the tenth of a milligram. This 
example shews how difficult it is to obtain a weight correctly 
to one part in a million. 

Numerical Exercise, 

A litre of water is to be weighed to the nearest milligram ; 
calculate how nearly you require to know the height of the 
barometer, the temperature of the balance case, and the specific 
gravity of the brass weights. 



4—2 



SECTION VIII. 

DETERMINATION OF THE DENSITY OF A SOLID. 

Apparatus required : Delicate balance, piece of quartz, 
specific gravity flask, air and water hath^, small pieces of 
quartz. 

The density of a substance at any point, is defined to be 
the quotient of the mass of a small volume of the substance at 
that point, by the volume. If the substance is homogeneous 
and m is the total mass, and v the total volume, we have 

p = m/v. 

Hence if the unit of mass is the gram, and the unit of 
length the centimetre, the density of a homogeneous body will 
be numerically equal to the mass of one cubic centimetre of 
the substance. 

In the metric system, the gram was originally chosen to be 
the mass of 1 cubic centimetre of water at its point of 
maximum density 3°*95 C. A kilogram, equal to 1,000 grams, 
deposited in Paris, serves as the ultimate standard for weights 
constructed on the metric system. Since the experiments 
determining the gram were made, however, physical instru- 
ments and methods of observation have improved, so that a 
small difference is now found to exist between the theoretical 
gram and the practical standard of mass. In consequence, the 
density of water at 3°*95 C. is not unity, as it was meant to be, 
but is 1 '000013. The difference is so small that it may gene- 
rally be neglected, but it would have to be taken into account. 



VIII DETERMINATION OF THE DENSITY OF A SOLID 53 

if for instance, 100 grams of water were to be weighed correctly 
to a milligram, and the volume occupied by the water calculated 
from the results to one part in 100,000. 

The specific gravity of a homogeneous substance, is 
defined to be the ratio of the mass of any volume of the sub- 
stance, to the mass of the same volume of water at 3°*95 C. 
As we may take the mass of 1 c.c. of water at 3°*95 C. to be 
unity, it is clear that the specific gravity is numerically equal 
to the density expressed in the C.G.s. system of units. This is 
an advantage, as in that system we may dispense altogether 
with the idea of "specific gravity" and always use that of 
" density " instead. Students must be clear, however, that the 
two terms are not synonymous, as the numerical value of the 
specific gravity is independent of the units of length and mass, 
while the number expressing the density will depend on those 
units. 

Since the direct determination of the volume of a body 
cannot be carried out accurately unless the body is of some 
regular shape, density determinations generally depend on a 
previous knowledge of the density of some standard substance, 
water being selected as the most convenient standard. 

Increase of temperature will in general diminish the density 
of a body, hence in stating the density, the temperature at which 
the number holds should always be specified. 

The density of water has been determined with great care, 
and has been found to decrease at an increasing rate per degree 
as the temperature rises. At 15° C, the ordinary temperature 
of the laboratory, the decrease of density of water for V C. is 
not much more than 15 parts in 100,000, while at 50° C. it is 6 
parts in 1,000. Tables giving the densities at various tempera- 
tures are given by Volkmann {Wied. Aim. xiv. p. 206; 1881), 
and Landolt and Bornstein (Physikalisch-Chemische Tahellen), 
but the second column of Table 32 (Lupton's Tables) will be 
sufficient for most purposes. 

There are a number of different methods by means of which 
the density of a body may be determined, and the best manner 
of proceeding in each case will depend on the available quantity 
of the substance, on its chemical properties and state of aggre- 



54 MECHANICS AND GENERAL PHYSICS VIII 

gation. We shall describe two methods, one of which will always 
be applicable if the substance is solid and insoluble in water. 
The method would have to be modified if the substance were 
soluble in water, or if it were hygroscopic. 

Method I. If a body is weighed first in air, and then 
suspended in a liquid, its apparent weight will be less in the 
second case than in the first, and it has been known since the 
time of Archimedes, that the apparent loss of weight is equal to 
the weight of the liquid displaced by the body. If M is the 
mass of the body, and p its density, its volume is M/p, and if 
(T is the density of the fluid in which it is weighed, the ap- 
parent decrease of mass will be 

M- , 
P 

The same holds for a weighing in air, the density \ of air 
being substituted for tr. 

Assuming the arms of the balance to be a and b cms. re- 
spectively, we have for the moments about the central knife- 
edge, of the forces on the two arms during the weighing in air, 
the quantities 

aMgll--") and hM,g(l-^y 

where ilf, is the apparent mass, and o-j the density of the weights. 
For equilibrium these moments must be equal, hence 

al/(l--)=61A(l-^ 

Similarly for the weighing in water, if ifg is the apparent mass 

Dividing the first equation by the difference between the 
first and second, we have 

a-\\ p) M^-M^' 
Or 



VIII DETERMINATION OF THE DENSITY OF A SOUD 55 

i.e., the excess of density of body over that of air 

= - . — ^ X (excess of density of liquid over that of air). 

It will be noticed that neither the inequality of the arms of 
the balance, nor the buoyancy effect of the air on the brass 
weights, enters into the result, so long as the weights are always 
placed in the same pan. This is due to density determinations 
depending on ratios of weights only. 

It will also be seen that the equation giving p corrected for 
the buoyancy of the air, may be obtained from the equation 

in which the air is neglected, by subtracting the density of the 
air from each density occurring in the equation. This may be 
seen on consideration to be due to the fact, that the weight of a 
body obtained in air would be the same as the weight obtained 
in vacuo, if the density of the body as it was transferred from 
air to vacuo, were decreased by the density of the air. This 
holds for both the quartz and the water in the above case, and 
the principle will be used in other cases. 

The numerical calculation is best carried out by writing 
1 — /c for <r, and p for MJ{M^ — M^ when we have 

p = p — p {k -^-X) -\-\ 

where the last two terms are small. 

Exercise I. Determination of the density of Quartz 
by weighing in water. 

1. Take two pieces of fine silk thread 40 cms. long, use 
one to tie round the quartz crystal provided, leaving a length 
of about 16 cms. with a loop at the end, hanging from the 
crystal. Cut away all unnecessary thread, and cut off equal 
lengths from the other piece. 

2. Suspend the quartz by means of the thread from the 
hook underneath the top of the support of the left-hand balance 
pan, place the other piece of thread in the right-hand jmn, and 
find the weights required to produce etiuilibrium. 



56 MECHANICS AND GENERAL PHYSICS VIII 

3. Remove the quartz from the balance, place it in water 
in a beaker, and boil the water to drive off the air bubbles 
adhering to the quartz. Then cool the water by pouring into 
it water from the tap gently without causing splashes. Place 
a small wooden stool across the left-band pan of the balance, so 
that the pan does not come into contact with it at any point. 
Support the beaker on this stool and suspend the quartz again 
from the hook. See that the quartz is entirely immersed in 
the water and that no bubbles of air adhere to it. Place a 
thermometer in the water, and note the temperature (Fig. 16). 
Cut off from the piece of thread in the right-hand pan a length 
equal to the length of thread in the water, and remove it. 
Weigh the quartz. 

Record as follows : — 

Density of Quartz. Method I. 

Date, Jan. 6th, 1893. 

Balance A. Box of Weights A. 

Weights in right-hand pan throughout. 

Temperature of Balance Case, 18°'4 C. 

Weight of Quartz in air (ifi) 40-882 grms. 

Temperature of Water, 20° C. 
Apparent Weight of Quartz in Water (Mg) ... 25*487 „ 

Loss of Weight = 15;395 „ 

M, 40-882 



M^-M, 15-394 
a = -9983 
X=-0012 



= 2-6556 



o-_X = -997l 

M 

^^^(.-X) = 2-6479 

X= -0012 



p = 2 6491 
Hence p the density of the quartz at 20° C. = 2*649 1. 




VIII DETERMINATION OF THE DENSITY OF A SOLID 57 

Method II. If the solid can only be obtained in small pieces, 
we may determine the density by the use of a 
"specific gravity flask " (Fig. 19), which is a small 
glass flask provided with a well-ground stopper 
traversed by a narrow channel. When it is filled 
with a liquid, and the stopper is inserted care- 
fully so as to exclude air bubbles, the excess of 
liquid will flow out through the capillary open- 
ing. By means of a piece of blotting-paper a 
small quantity of the liquid may be removed, so 
that it just reaches to a marked height in the 
opening. The flask may in this way be repeatedly filled to the 
same level, and if its temperature is the same, the volume of 
its contents will be the same. 

The flask having been cleaned and dried, the density re- 
quired is determined by the following series of weighings, the 
letters, F, &c., representing the weights obtained : — 

1. The flask dry, F. 

2. The flask dry with the dry solid placed inside, F-\- M^. 

3. The flask with the solid inside, after filling up to the 

mark with a liquid of known density a^ at a tempera- 
ture ^, F-\-M^-\- W^. 

4. The flask entirely filled up to the mark with a liquid 

of density Cj at a temperature f.^, F ■\- W^. 

Since a^ is the density of the liquid at fj, the volume of the 

W 

flask at ^3, neglecting the effect of the air on the weighing, is — - , 

and the volume at ti will be — ^ ^ - ^ , where a is the coefl^cient 

<r, 1 + orfg 

of cubical expansion of glass. Similarly the volume occupied 

by the liquid at <i = . The difference between these volumes 

is the volume occupied by the solid at ^, and the density p of 
the solid at t, is the mass divided by this volume. 

The effect of the buoyancy of the air may be taken into 



58 MECHANICS AND GENERAL PHYSICS VIII 

account by subtracting the density of air X, from each of the 
densities in the equation for p (page 55), and we thus get 

P-^= W„. l + at. tV. • 

cTo — X ' 1 4- olL o-j — \ 

If we take weighings of the flask empty and when filled with 
each of two liquids, we may compare the densities of the liquids, 
since on making M^ = in the above equation, we have 

W^ 1 + a^ ^ Tf 1 
0-2 — X ' 1 4- a^o cTi — \ ' 

an equation from which we can determine one density if the 
other is known. 

If in the former equation, we take the liquid to be water in 
each case, so that cr^ and a^ are both nearly unity, we have, 
using the methods of approximation given in Intermediate 
Practical Physics (page 16), 

P — \= == ^ . ( <Ji — A, ). 

^ W,(l + a;-a,-hOit,-t,)-W, 

Or writing o-j = 1 — /c, and 

^ W^(l-\-<r^-(r^-{-at,-t^)-W^' 
we have 

p = p - p (/Ci + X) + X. 

Exercise II. Determination of the density of Quartz 
by the specific gravity flask method. 

1. Clean the 50 gram flask (Fig. 19) provided, by washing 
it if necessary with a strong solution of caustic potash, and then 
with water from the tap. The potash is to be thoroughly 
removed with tap water, and the final washing made with dis- 
tilled water. 

Place the flask on the shelf of an air bath kept at about 
120° C, insert a glass tube into the flask, through an opening 
in the top of the bath. Look at the flask occasionally, and 
when the drops of water have evaporated from the sides, draw 
the moist air out of the flask, by applying the mouth to the 



VIII DETERMINATION OF THE DENSITY OF A SOLID 59 

Upper end of the glass tube. Repeat this several times, then 
remove the flask and allow it to cool. If no moisture is de- 
posited on its inside surface as it cools, place it in the left-hand 
pan of the balance, and weigh. 

With the quantities of the substance available, it will be 
suflScient to weigh accurately to the nearest milligram, and the 
process of weighing may therefore be shortened by taking only 
three readings of the vibrating pointer (two on one side and 
one on the other), instead of five, as in the previous exercises. 

2. Dry thoroughly 15 or 20 grams of the broken up pieces 
of quartz with which you are provided, and place in the flask. 
Weigh the flask and contents, and hence deduce the weight of 
the quartz. 

3. Pour some distilled water into the flask. If air bubbles 
adhere to the small pieces of quartz, it will be necessary to 
expel them. For this purpose place the flask, with the solid 
pieces completely covered with water, in the air bath, and heat 
up carefully until the water just begins to boil. Take out the 
flask, and after allowing a little time for cooling, fill up with 
distilled water, and then place it in the water bath with 
which you are provided, keeping it in position by indiarubber 
bands. 

Stir the water well, and keep it at a temperature two or 
three degrees higher than that of the room. 

Read the temperature to 0°05 C. 

The temperature of the bath must be kept constant, and 
the flask kept in the bath for a sufficient length of time to 
allow the water in it to take up the same temperature. This 
may be ascertained by placing a thermometer in the flask. 

The stopper is then inserted, and the quantity of water 
in the flask is adjusted so that it reaches to the mark across 
the capillary opening in the stopper. 

Take the flask out of the water and dry the outside ctxrefully, 
taking care to avoid heating it by contact with the hand and 
forcing out any of the water. The object of filling the flask 
with water at a temperature above that of the air is now 
apparent, for when the flask is taken out of the water bath, 
the water in it will contract, and thus the danger of losing any 



60 



MECHANICS AND GENERAL PHYSICS 



VIII 



by accidental heating of the flask is greatly diminished. After 
the flask has been dried on the outside, leave it in the balance 
case for a few minutes so that it may acquire the temperature 
of the balance case, then weigh again. 

If the flask is not perfectly dry outside, evaporation will 
take place and the Aveight of the flask will slowly diminish. 
Ascertain that the weight remains constant. 

The difference between the weights of the flask in this and 
the previous weighing, will give the weight W^ of the water it 
contains. 

4. Now take out the quartz, fill the flask with distilled 
water and place it in the water bath. Keep the temperature 
of the bath nearly the same as it was in the second weighing ; 
a correction will be necessary if any difference exists between 
the two temperatures, which should be known to the twentieth 
of a degree. Close the flask, dry and weigh it. 

Record as follows : — 

Density of Quartz. Method II. 

Date, Jan. 6th, 1893. 

Balance A. Weights A. Flask 13. 

Weights in right-hand pan throughout. 

Temperature of Balance Case, 16"'0 C. 



Weight of flask, i^ 

Weight of flask and quartz, F-\-Mi 
Hence weight of quartz, M^ 

Weight of flask, quartz, and water) 
Sitt,= lS°,F-\-M,+ Wr J 

Hence weight of water at 18°, Wi 

Weight of flask filled with water [ 
8itt,=^19°\F+W^ J 

Hence weight of water filling flask ) 
at 19°, W, J 



17-325 grms. 
32012 „ 



14-687 



= 76-438 



= 44-426 



67-274 



49-949 



1 This temperature differs more from t^ than it need be allowed to do. The 
difiference is taken great here to shew clearly the magnitude of the correction 
introduced. 



VIII DETERMINATION OF THE DENSITY OF A SOLID 61 

At 18^ <T, = -99867 
At 19°, Co = -99847 



o-i-o-., = -00020 a =000023 
a{t,-Q = -00002 

Sum = 00022 



IT, (o-j — tTa + a^ — tj) = Oil grms. 



W^il-^a.-a. + aU-U) = 49-960 

Fi = 44426 



Tr,(l + <r,-o-2 + a^a-^)-Trx = 5-534 

"-■^ - ^ 

K, = -00133 
X = -00122 



K^ + \ = -00255 

/)i(/tfi + X) = 0067 

X = 0012 



Hence density p of the quartz at 18° C. = 2*6485 



SECTION IX. 

DETERMINATION OF THE DENSITY OF A LIQUID. 

Apparatus required : Balance, specific gravity flask, 
salt solution, water bath, Mohr's balance, and hydrometer. 

Method I. By the specific gravity flask. 

In the previous section (p. 58) it has been shewn how the 
density of a liquid may be found by means of the specific 
gravity flask, and we now proceed to apply the method to the 
determination of the density of the salt solution provided. 

Make use of the specific gravity flask used in the deter- 
mination of the density of quartz. The apparent weight of the 
flask empty was found to be F grams, and the apparent weight 
of water filling it at the temperature ^i, TTj grams. 

The flask should be dried, filled with salt solution, and 
placed in the water bath at a temperature 2 or 3 degrees 
higher than that of the balance case. 

After allowing the solution to take up the temperature of 

the bath, put in the stopper and carefully remove with filter 

paper the drop at the top of the hole in the stopper. Let the 

observed temperature of the water bath be fg- Remove the 

flask, dry the outside and weigh. Let TTg be the apparent 

weight of the salt solution. 

W 
The volume of the flask at the temperature ^i is — ^ when 

<7i is the density of water at that temperature ; the volume of 

W 
the flask at 0° C. will therefore be — ^ (1 + a^i), where a is the 



IX DETERMINATION OF THE DENSITY OF A LIQUID 63 

coefficient of expansion of glass, and this volume must be the 
same as that calculated from the corresponding weighing of the 

liquid, i.e. - " (1 + a^). The correction of the weighings for 

buoyancy is introduced by writing o-, — \ for a^ and a^ — X for 
o-j (see page 55), hence 

(Ta — X 1 + af 1 * cTi — \ ' 
Or since a^i, aL are both small 



which equation serves to calculate a^ if o-j is known. 
Record as follows : — 

Density of Salt Solution. Method I. 

Date, Jan. 8th, 1893. 

Balance A. Weights A. Flask 13. 

Weights in right-hand pan throughout. 

Temperature of Balance Case, 18° C. 

Weight of dry flask, F = 17-325 grams 

„ flask filled with water at 19° = 67-274 .. 



„ water filling flask at 19° = TTj = 49949 

„ flask filled with salt solution at 19°-2 = 70-282 

„ salt solution filling flask at 19°-2 = F, = 52*957 
Density of water at 19° = a, = '9985 
„ of air at 18° =\ = 0012 

.'. <7,-\ ... = -9973 

a for glass = 000023, .-. o^ - <2= 0. 

tf;'4T949 = ^^^^^ 

<r,-X= 1-0601 X -9973 ... = 10572 
X = 0012 

.•.o-a=!density of solution ... = 10584 



64 



MECHANICS AND GENERAL PHYSICS 



IX 




Fig. 20. 



Method II. By Mohr's Balance. 

Mohr's balance (Fig. 20) is an ordinary balance modified so 
as to enable determinations of 
densities of liquids to be made 
rapidly. One arm of the balance 
is divided into 10 equal parts, 
and carries, suspended from its 
end by a fine silk fibre, a glass 
thermometer which is immersed 
in and indicates the temperature 
of the liquid. The other arm 
carries a counterpoise, the end of 
which is pointed, and comes close 
to a corresponding pointer on the 
frame of the balance. The balance 
can be clamped at any height by means of the screw in the stem. 

Place the balance on the stand in such a way that the 
levelling screw in the stand lies in the vertical plane through 
the beam. 

Suspend the thermometer from the graduated arm, and 
turn the levelling screw till the pointers are in line with each 
other. Fill the small test tube with water at 15° C. Raise 
the balance by means of the expanding stem, place the test 
tube under the thermometer, and lower the balance till the 
thermometer is entirely immersed in the water. The balance 
will no longer be in equilibrium owing to the upward force 
of the water on the thermometer, which we have seen is equal 
to the weight of the water displaced by the thermometer. To 
produce equilibrium weights must be placed in the notches of 
the arm of the balance. 

The largest brass weights provided are equal, the other 
weights are -^ and j^ respectively of the largest weights. 

The notches are marked from the centre of the arm to the 

end 1, 2, 3 9, the hook under the end of the arm being at 

the tenth notch. If we call the weight of the largest brass 
pieces 1, then if 1 is placed in the notch 6 it is equivalent to a 
weight of '6 placed at the end, and so on. Thus the weight 



IX DETERMINATION OF THE DENSITY OF A LIQUID 65 

necessary to produce equilibrium is given by the readings of 
the notches in which the weights in order of magnitude are 
placed, the unit in which the weight is expressed being that of 
the largest brass weight. The weight found is in terms of this 
unit that of a volume of water at 15° equal to the volume of 
the thermometer. 

Remove the weights, raise the balance, and after drying the 
test tube and the thermometer replace the water by the liquid 
the density of which is required and see that the thermometer 
in the liquid indicates again 15° C. 

Again determine the weight to produce equilibrium. Since 
this is the weight of a volume of the liquid equal to that of the 
thermometer, the ratio of the density of the liquid at 15° to 
that of water at 15° is the ratio of the two weights. The 
density of the liquid at 15° is therefore the product of this ratio 
and the density of water at 15° C. (= -999). 

The weights are generally arranged so that one of the 
heaviest will when hung in the hook produce equilibrium 
when the thermometer is in water. In this case the density of 
the liquid can, to within 001, be read off immediately from 
the positions of the weights when the thermometer is in the 
liquid. 

Method III. By the Hydrometer. 

When a solid floats on a liquid it is acted on by two forces, 
one its weight downwards, and the other the pressure of the 
liquid upwards. Since the body is in equilibrium these two 
forces must be equal. But the pressure of the liquid is equal 
to the weight of the liquid displaced. Hence if W be the mass 
of the solid, V the volume of liquid displaced, <r its density, we 
must have W = Va, or 

(7 

Hence when the solid floats on a denser liquid it sinks less 
than it does in a lighter liquid and the volume immersed varies 
inversely as the density of the liquid. 

8. p. 6 



66 MECHANICS AND GENERAL PHYSICS IX 

The Hydrometer provided (Fig. 21) has been 
graduated by being placed in liquids the densities of 
which had been determined by the previous methods. 
Hence to get the density of a liquid it is simply 
necessary to place the hydrometer in it and read the 
position of the surface of the liquid on the graduated 
stem. 

A test tube mounted on a block of wood is provided 
for holding the liquid. To read the scale, hold the eye 
below the level of the surface of the liquid, and gradu- 
ally raise it till that surface is seen foreshortened into a J\ 
straight line. The position of this line on the scale is 
the density required. 

Collect your results as follows : — 

Density of salt solution by flask ... ... 1'0584 \y 

„ „ „ Mohr's balance ... 1'059 ^^ 

hydrometer ... 1-058 Fig. 21. 



SECTION X. 

MOMENTS OF INERTIA. 

The following pages contain a short statement of the prin- 
cipal propositions concerning Moments of Inertia and the 
Compound Pendulum. Students should read them carefully 
and work out the examples before proceeding to the practical 
exercises on Moments of Inertia and the Pendulum. 

Some of the propositions are given without proof, and the 
students are referred to books on Dynamics {e.g. Worthington, 
Dynamics of Rotation ; Hicks, Elementary Dynamics) for a more 
detailed treatment. 

The Moment of Inertia of a particle about an axis is defined 
as 7nr^, where m is the mass of a particle and r the perpendicular 
distance between the particle and the axis. 

The Moment of Inertia of a number of particles about an 
axis is the sum of the Moment of Inertia of each. 

Thus particles 7/1,, m,, m,, at distances r^^r^, ?•,, r^, from an 
axis have a moment of inertia about that axis equal to 

niirx^ + nu^r^* + w,r,' + . . . . 

From this it follows that the moment of inertia of a body 
about any axis is equal to the sum of the moments of inertia of 
the separate parts of the body about the same axis, it can there- 
fore be found by dividing the body into a number of small parts 
and adding the products of the masses of the parts into the 
squares of their perpendicular distances from the axis. 

5—2 



68 MECHANICS AND GENERAL PHYSICS X 

The following moments of inertia are frequently required: — 

M always denotes the mass of the body which is supposed to 
be of uniform density. 

I. A right solid parallelepiped, whose edges 

are 2a, 26, 2c, about an axis through its centre m^^'^^^ 
perpendicular to the plane containing the edges 3 

b and c. 

II. A solid cylinder of radius r about its ^r* 
axis of figure. 2 

III. A solid cylinder of length 2a and radius 

r about an axis through its centre perpendicular M ( — -^'—j 
to the length of the cylinder. 

IV. A sphere of radius a about any axis ^ 2a^ 
through its centre. 5 

The radius of gyration of a body with respect to an axis is 
the distance at which a particle of the same mass as the body 
would have to be placed in order to have the same moment of 
inertia. Hence, if the Moment of Inertia is written in the 
form Mk^, M being the mass of the body, k will be the radius 
of gyration. 

The moment of inertia of a body about any axis is equal to 
the moment of inertia about a parallel axis through its centre 
of mass, together with the product of the mass of the body and 
the square of the distance between the axes. 

The theorem of p. 67 enables us to calculate the moment of 
inertia of bodies from which parts have been removed, if the 
moments of the complete bodies and of the removed parts are 
known separately. 

Thus let it be required to find the moment of inertia of a 
circular ring about an axis passing through its centre and per- 
pendicular to the plane of the ring. 

Let the inner and outer radii of the ring be r^ and r^, and 
imagine the inner space to be filled with matter of the same 
density so as to convert the ring into a disc. Let rrii be the 



X MOMENTS OF INERTIA 69 

mass of the smaller disc of radius 7\, which fits into the circular 
hollow of the ring, and let m^ be the mass of the complete disc 
of radius r,. Then from the proposition, if / is the unknown 
moment of the ring 

/ + __== ^i,^^, 

or /= 2 • 

But if p is the density of the ring 

T/ii = irpr-^ and m.^ = irpr^. 
Hence 

If 3/ is the mass of the ring M = irp {r^ — r^% hence finally 

Exetxises. 

1. Find the moment of inertia of a solid parallelepiped 
about an edge. 

2. Find the moment of inertia of a cube with a central 
spherical hollow about an edge of the cube. 

3. The moment of inertia of a hollow circular cylinder is 
required about an axis at right angles to the length of the 
cylinder and passing through one of its end planes. 

The moments of inertia of solid bodies are of importance as 
these quantities often occur in physical problems. Thus the 
Kinetic Energy of a body rotating about an axis is expressed 
by \Iw*, when / is the moment of inertia of the body about the 
axis, and w is the angular velocity. 

If a body oscillates about an axis under the influence of ex- 
ternal forces, as for example a pendulum under the influence of 
gravity, the time of vibration depends on the moment of inertia. 
Oscillations like those of a pendulum occur whenever a body is 
capable of rotating about an axis, and is acted on by forces 



\ 



70 BfECHANICS AND GENERAL PHYSICS X 

such that if the body is disturbed from its position of equili- 
brium, a couple acts on the body tending to bring it back. If 
the couple is proportional to the angle through which the body 
is turned, the oscillations are strictly isochronous, that is, the 
time of vibration is independent of the amplitude of oscillation. 
This is only approximately true in the case of the pendulum, 
but much more nearly true if the body is capable of oscillating 
in a horizontal plane, being brought back to its position of 
equilibrium by the torsional forces of a wire by which the 
body is suspended. The time of vibration is then given by 
the equation 

3^ = 2^5^' 

where / is the moment of inertia of the body about the axis 
around which it turns, and N is the couple per unit angle of 
deflection, tending to bring the body back. 



SECTION XI. 

EXPERIMENTAL DETERMINATION OF MOMENTS OF 

INERTIA. 

Apparatus required : Rectangular metal block, fine loire, 
and hi filar suppoi'ts. 

The moment of inertia of a body about an axis through its 
centre of gravity, may be found experimentally by suspending 
the body, either by a single fibre or by a double one, in such a 
way that it can perform torsional oscillations about that axis, 
and determining the time of an oscillation. A second body, the 
moment of inertia of which, about some axis through its centre 
of gravity is known, either on account of its regular shape or by 
previous determination, is then attached to the first, in such a 
way that the axis is coincident with that of the fibre, and the 
time of an oscillation again determined. From these times 
the moment of inertia of the first body can be found in terras 
of that of the second by the equation : /„ = I-^Tq^JT-^ — Tq^, where 
Ii is the moment of inertia of the mass added, To and Ti are the 
times of oscillation. This is the method often used when the 
moment of inertia of a suspended magnet is to be determined. 

If the bifilar method of suspension is adopted, the body, the 
moment of which is required, is laid in a stirrup at the lower 
end of the suspension, with the axis about which the moment 
is required vertical and half-way between the two suspending 
fibres. If it is rotated in a horizontal plane through a small 
angle, the suspension resists the rotation with a force propor- 
tional to the sine of the angle of twist. 

The couple N for small angles of rotation being = Mg ~j ' , 

we obtain the time of vibration by substituting this value in 
the general equation 

where M = mass of body suspended. 



72 



MECHANICS AND GENERAL PHYSICS 



XI 



I = moment of inertia of body suspended about axis of 

suspension. 
I = length of each fibre. 
di,di = distances apart at top and bottom. 

g = gravitational acceleration. 
Since / = Mk^ where k is the radius of gyration of the body 
about the axis we have for a small oscillation 



V gdido' 




Exercise I. To verify the relation between the time of 
oscillation and the constants of the bifilar suspension. 

Suspend from the lower end of the fibres the rectangular 
block provided (Fig. 22), passing the fibre 
through the holes at the ends of the small 
brass strip which can be screwed to the 
block, then over the larger pulley above 
the separated horizontal clamps forming 
the upper support. When the block has 
come to rest, screw up the clamps so that 
the two fibres are held firmly, then set the 
block in oscillation through an arc of about 
20°, and determine the time of oscillation 
and the mean arc of twist on each side 
of the equilibrium position (Section III). 
Measure the length of the fibres and their 
distances apart at the top and bottom. 

Now unscrew the clamp, pass the fibre 
through the holes nearer the centre of the 
strip, and over the smaller pulley, then 
clamp it, measure the length of the fibres 
and the distances apart at top and bottom, 
and again determine the time of vibration. 

Reduce the length of the fibres by 
moviog the clamp to a lower part of the 
support and again clamping the fibres. 
Again determine the time, and measure the 
distances apart of the fibres. 




Fig. 22. 

lengths and 



XI EXPERIMENTAL DETERMINATION OF MOMENTS OF INERTIA 73 



Tabulate the observations and results as follows :- 
3 Jan., 1898. Apparatus A. 



Length 
cms. 


Distances apart 
cms. 


did. 


^ 


Axis of 

block 

vertical 


Time 
seconds 


(time)5 


^. 


Top 


Bottom 


76-0 
75-7 
41-5 


3-48 
1-50 
1-50 


3-53 
1-40 
1-40 


12-28 -161 
2-10 -0277 
2-10 ; -0506 


longest 


2-927 
6-960 
5-201 


8-57 
48-35 
27-00 


1-38 
1-34 
1-37 



The constancy of the numbers in the last column verifies 
the law for the bifilar suspension. 

Exercise II. To determine a Moment of Inertia. 

The rectangular metal block provided may be attached to 
the bifilar suspension, so that its three principal axes coincide 
in turn with the vertical axis of the suspension. 

The previous observations suffice to determine the moment 
of inertia about its longest axis. 

Attach the block to the suspension so that the axis of mean 
length is vertical, and determine the time of oscillation, using 
the full length of fibre and maximum distance apart. 

Next attach the block with its shortest axis vertical and 
determine the time. 

Calculate from the observation the moments about the three 
principal axes, arranging the work as follows : — 

Block marked A. 
3 Jan., 1898. 
I = 760 cms. 



d, = 3-48) 
rf, = 3-53 



did, 
I 



= 161, 






= 40-2. 



Axis 
vertical 


Time 
see. 


f« 


At* 


4ir« Al 


I 
observed 


I 
calculated 


long 

mean 

short 


2-927 
4-33 

5-128 


8-57 
18-49 
26-30 


•21 
•46 
•65 


8-6 
18-6 
•26-6 


1026 1010 
2076 2056 
3120 3182 



The fifth column gives the values of K 



74 MECHANICS AND GENERAL PHYSICS XI 

Weigh the metal block, multiply each value of k- by the 
mass to get the moments of inertia, and tabulate as in the sixth 
column above. 

Take the dimensions of the block, and calculate the moments 
by the formula 



where a, 6, c are the half lengths of the sides of the block. 
Record as follows and enter in preceding table. 

jlf= 1200 grams 

a = 7-5 cms. .'. a^ = bQ-^ .\ ia = 1010 

6 = 5-5 62 = 3Q.2 /j, = 2056 

c = -55 c2= -3 /o = :3182 

The oscillations of a bifilar suspension are like those of a 
pendulum only approximately isochronous, and the arc of 
rotation should therefore be small if great accuracy is required. 
If the arcs are large the correction given in the next exercise 
may be applied. 



SECTION XII. 

THE COMPOUND PENDULUM. 



A Compound Pendulum is one in which the mass is distri- 
buted over a finite volume and not concentrated at a single 
point as in the simple pendulum. Strictly speaking every 
pendulum is a compound pendulum, and we can only approach 
the ideal simple pendulum by reducing the 
volume of the heavy bob of an actual pendulum 
as much as possible. 

In Fig. 23 let G be the centre of mass of a 
heavy body of mass in capable of turning about 
an axis through at right angles to the plane 
of the paper. 

In the position of equilibrium, the centre of 
mass G will be vertically under 0. If the body 
is displaced through an angle 6, the resultant 
gravitational force passing through G, will have 
a moment mgh sin 6 about 0, g being the ac- 
celeration of g^vity and h the distance OG 
between the centre of mass and the axis of rotation. This 
couple is proportional to sin 6, which if 6 is small will be nearly 
equal to 6 in circular measure, and the couple tending to bring 
the body back into its position of rest, will be nearly mghS, i.e. 
nearly proportional to the angle of displacement. Under these 
circumstances the body will o.scillate about the point 0, and the 
time of oscillation will for small displacements be independent 
of the amount of the displacement, i.e. the oscillations will be 
isochronous. 




Fig. 28. 



76 MECHANICS AND GENERAL PHYSICS XII 

The time of oscillation will be 

V mgh' 

where / is the moment of inertia of the pendulum about the 
axis of oscillation, and mgh the couple for a displacement of a 
right angle. 

If K is the radius of gyration of the body about 0, I = mK^ 
and the equation becomes 

Definition. The length of the equivalent simple pendulum 
is the length of the simple pendulum having the same time of 
oscillation as the given body. 

As a simple pendulum of length I has a time of oscillation 

equal to 27r . / - , it follows that if I be the length of the equi- 

V if 

valent simple pendulum 

I = K'lh. 

If m^'2 be the moment of inertia of the compound pendulum 
about an axis parallel to the axis of oscillation, and passing 
through the centre of mass G, we have 

K^^k' + hK 
Hence 

i.e. (I -h)h = k\ 

If a point P be taken in the line 00, Fig. 23, such that 
OP is equal to the length I of the equivalent simple pendulum, 
P is called the centre of oscillation, while is called the centre 
of suspension. 

Hence the radius of gyration with respect to the centre 
of mass, is the geometrical mean between the distances of the 
centre of mass from the centres of suspension and oscillation 
respectively. 

If the body is suspended from P, will become the centre 
of oscillation, for k^ being a constant, h and l — h may be inter- 
changed without interfering with the truth of the equation. 



XII 



THE COMPOUND PENDULUM 



77 



Let a pendulum be constructed, Fig. 24, such that it is 
capable of vibrating about either of two knife-edges 
A and B, the line joining the edges passing through 
G the centre of gravity of the pendulum. If the 
knife-edges can be adjusted so that the time of oscil- 
lation is the same whether the body oscillate about A 
or about B, then unless A and B are equidistant from 
G, the length AB is that of the equivalent simple 
pendulum, and as that distance can be accurately 
measured the value of g can be found accurately from 
the equation 




Fig. 24. 



This is the principle of Kater's Pendulum. 

The isochronism of the pendulum has been stated to depend 
on the equality of sin Q and 6 for small values of 0. If Q be- 
comes large the quantities are no longer equal, and the time of 
oscillation will vary with the angle of displacement. In that 
case a closer approximation is given by 



^-^Vk.^S). 



where Q is the semi-arc of oscillation. 

Hence if T,^ is the time of an infinitely small oscillation and 

T the observed time we may write ^0 = ^(1 — ^^). The 

following table gives the value of ^/16 for different arcs of 
swing. 



6 


^/16 


r 

10- 
20' 


•000019 
•000476 
•001904 
•007615 



In general there are four possible centres of oscillation on 



78 



MECHANICS AND GENERAL PHYSICS 



XII 



every straight line, such that the times of oscillation of the body 
about them are equal. 

Let A be any point of the body, G the centre of mass, and 
let the axis of rotation be at right angles to the plane of the 
paper (Fig. 25). With G as 
centre, draw a circle through A, 
join AG and produce to cut the 
circle again in A' . Then since 
GA' = GA the time of oscillation 
about A' will be the same as 
that about A. If AG = h and k 
is the radius of gyration about 
the axis passing through G, the 
length of the equivalent pendu- 
lum will be {k^-{-h^)/h. Make 
AB equal to this length. Then 
if the centre of the suspension is 
at B, the time of oscillation is 
the same as before, and therefore the same also for any point 
B' on the circle passing through B and having G as centre. 
Two circles may therefore be drawn such that the times of 
oscillation about all points of them are the same. A straight 
line such as CG\ not necessarily passing through G, will inter- 
sect these circles in four points, about which the times of oscilla- 
tion will be equal. 

Exercise. Calculate the length of the equivalent simple 
pendulum of: — 

1. A brass rod about its ends. 

2. A brass rod about a point half-way between its ends and 
its centre. 

3. A sphere about a tangent. 

4. A cube about one of its edges. 

5. A cylinder about one of its generating lines. 




SECTION XIII. 

EXPERIMENTAL DETERMINATION OF THE 
EQUIVALENT SIMPLE PENDULUM. 

Apparatus required : Brass bar with sliding knife-edges , 
support, and simple pendulum. 

A brass bar, in one end of which a number of holes have 
been drilled in order to make it unsymmetrical, is 
provided (Fig. 26). On it slides a knife-edge, so 
that it can be set swinging about an axis at right 
angles to its length at a distance from its centre 
of gravity which can be varied. 

In front of this brass bar a leaden ball is sus- 
pended by means of a thread, the length of which 
is adjustable. 

The experiment consists in adjusting the length 
of this simple pendulum till it has the same time 
of oscillation as the compound pendulum. 

Place the knife-edge about a cm. from one end 
of the bar and suspend the bar in its support. 
Make the thread of the simple pendulum about 
I the length of the bar, push both pendulums aside 
with the palm of one hand, and suddenly withdraw 
the hand so that the pendulums start simul- 
taneously. Watch the oscillations of both pendu- 
lums, noting which of the two gains on the other. 
If after a few oscillations one is decidedly ahead 
of the other, the simple pendulum must be length- 
ened if it swings too quickly, and shortened if it '** 
swings' too slowly. After a few trials the length may be 
adjusted so as to make the times of oscillation of the two 
pendulums equal to each other to within 2 or 3 per cent. 

When that is the case, set the pendulums swinging together 
once more, count the oscillations of the bar, noting again 



80 MECHANICS AND GENERAL PHYSICS XIII 

which pendulum gains on the other; go on counting the 
oscillations until the pendulums swing again together. 

If n is the number of oscillations of the compound pendulum 
counted, the simple pendulum will have performed (?i — 1) or 
(7i-f 1) oscillations, according as it is too long or too short. 
The number n should not be less than 30, otherwise a readjust- 
ment of the simple pendulum should be made. If one pendulum 
gains so slowly on the other that the number n cannot be fixed 
to one or two units, find ?ii and n^, so that i^ is the number at 
which it begins to be doubtful whether the pendulums swing 
together, and iu the number at which it begins to be certain 
that they have ceased to swing together; n will then be very 
nearly equal to (?ii + n^l% 

If V is the actual length of the simple pendulum, the length 
I of the simple pendulum which would swing exactly in the 

same time as the brass bar is equal to T f 1 j or T ( 1 + J , 

according as the simple pendulum made ?i — 1 or ?2 + 1 oscilla- 
tions, while the compound pendulum made n. 

If k is the radius of gyration of the bar about its centre of 
mass (r, and h the distance of the centre of mass 
from the point of suspension 0, then, as shown in 
the previous section, PO x OG = k^, which may 
therefore be determined from the observations 
made. 

Measure the distance of the knife-edge from 
the top of the knife-edge clamp, and the distance 
from the top of the clamp to the upper end of the 
bar. The sum of these is the distance -4 of the 
knife-edge from the top of the bar. 

Five positions of the knife-edge should be taken, 
about 1, 10, 20, 30, and 35 cms. from the top of 
the bar ; the bar must then be reversed and five ^^8- ^7. 
more observations taken. 

Determine the centre of mass G of the bar by removing the 
knife-edges and balancing the bar on one of them. 

Calculate for each experiment the distance OG. From this 
and the measurement of I calculate PG. 



XIII EQUIVALENT SIMPLE PENDULUM 

Enter as follows : — 

21 Jan., 1894. 
Bar a. 
Distance of centre of mass from solid end of bar = 50*8 cms. 
„ from top of clamp to knife-edge ... = '9 „ 

Solid End of Bar at Top. 



81 



Distance 

top of 

bar to 

top of 

! clamp 


Distance 
from top 
to knife- 


AG-AO 
= 0O 


n 


1 

Simple ' 

pendulum 1 

obeerred I 

I' 1 

1 


n-l 
or 

n-fl 


l=OP 


OP-OO 
=P0 


00.P0 


cm. 


cm. 


cm. 




cm. 1 




cm. 


cm. 




•7 


1-6 


49-2 


38 


63-8 


39 


67-2 


180 


885 


8-2 


9-1 


41-7 


36 


59-5 1 


37 


62-8 


2M 


879 


18-6 


19-5 


31-3 


33 


55-8 


34 


59-2 


27-9 


873 


29-8 


30-7 


201 


39 


60-2 


40 


63-3 


43-2 


868 


34-4 


35-3 


15-5 


33 


6G-5 , 


34 


70-6 


55-1 


854 


Mean 872 



Draw up a similar Table of results when the solid end is at 
the bottom. 

Draw a curve to express the above results, taking distances 
of the point of suspension from one end of the bar as abscissae, 
and the lengths of the equivalent simple pendulums for the 
points as ordinates, as shewn below : — 



= T == =EE::;:::::::::: ::::: 


■± :::::: 


— :::± .:_. 


- "- :± 




_., :1 _-- 


::::::::::2:::: ;::::::::;!::: 


: .i^::-_,z . ^. . _ ^! 


9T »• "* 1. ^ . ,- "^ 




M 








• 




::!::::::""::::::"":::"::::: 








::8::::3£:::i E2ii:2E:::] a:::: 


:::J::::::::ffi:::: :l :: 


1 tit 1 



Fig. 28. 



8. P. 



SECTION XIV. 

DETERMINATION OF YOUNG'S MODULUS BY THE 
BENDING OF BEAMS. 

Apparatus required : Uniform wooden beam and supports, 
weights, silvered glass scale and support. 

When a beam of breadth b and height h, supported at two 
points I apart, carries a mass ^ at a point half-way between 
the supports, this point is depressed by an amount d given by 
the equation 

where e is Young's modulus for the material of the beam, and g 
the gravitational acceleration. 

To verify this relation, place the two supports provided, so 
that the top edges are in the same horizontal plane and about 
80 cms. apart. Look across them to see that they are parallel. 

Place the wooden beam so that the marks near its ends 
come over the supports, and behind its middle point place a 
graduated mirror with the scale vertical (Fig. 29). 

Arrange the rod so that its greater breadth is horizontal, 
and read on the scale the height of its middle point. 

By means of a hook suspend a 500 gram weight from the 
middle of the beam, and read on the scale the depression of the 
beam. Increase the weight to 1000 grams, and again read the 
position of the middle point. Verify, by means of your obser- 
vations, that the deflection is proportional to the deflecting 
weight, taking into account the weight of the hook. 



XIV 



YOUNGS MODULUS BY BENDING 



83 



Turn the beam so that its greater breadth is vertical, and 
repeat the observations. 



^ 






Fig. 29. 

Use a shorter length of the beam, and with the greater 
breadth horizontal repeat the observations. 

From the observations calculate, using the above formula, 
Young s modulus for the material of the rod. 

Arrange your results as follows : — 

3 Feb., 1899. 

Beam of Oak Wood marked 0. 3. 

Length of beam between supports ... =79*8 cms. 

Greater breadth ... ... ...= 1*36 cms. 

Lesser breadth ... ... ... = '88 cm. 

Greater breadth horizontal. 



Load in 
grams 


Reading 
at centre 


Depression 
cms. 


Depression 
per gram 





7-90 






509 


6-65 


1-25 


•00246 





7-90 






963 


5-55 


2-45 


•00254 





7-90 










Mean 


•00250 



6—2 



84 



MECHANICS AND GENERAL PHYSICS 



XIV 



1 981 



(79-8)3 



4 '0025 1-36 X (•88)' 



= 5'38 X 10*" dynes per sq. cm. 



Similarly for the observations taken with the greater 
breadth vertical and with the shorter length of beam, the 
values of e being collected as follows: — 



Oak Beam, 0-3. 


e 

in dynes 

per sq. cm. 


Greater breadth, horizontal 
„ „ vertical 
Shorter beam 


5-38 X lO'** 

5-24 

5-29 



Experiments with the shortened length of beam are here 
introduced in order to illustrate the law according to which the 
deflection is proportional to the cube of the length. If the 
sole purpose of the exercise had been to determine Young's 
modulus, these experiments would not have been necessary. 

Place the beam in its first position, and after observing the 
reading on the scale, place the 500 grams weight at a point 
which divides the beam in the ratio 1 : 3. Read the deflection 
at the middle point. Increase the weight to 1000 grams, and 
again read the deflection. 

Now place the mirror behind the point at which the weight 
has been applied, and remove the weight to the middle. Read 
the deflections for 500 and for 1000 grams, enter as below and 



Load 


Beading 


Deflection 
cms. 


Oat A 

500 gr. at A 

lOOOgr. at A 


7-85 at B 
6-70 at B 
5-65 at B 


115 at B 
2-20 at B 


OatB 

500 gr. at B 

1000 gr. at B 


7-55 at A 
6-40 at A 
5-40 at A 


M5at A 
2-15 at A 



XIV young's modulus by bending 85 

verify that they are the same as those found in the former case. 
Hence the deflections at B due to a weight at A is equal to the 
deflection at A due to the same weight at B. 

If metal bars are used the deflections will be smaller than 
in the case of wood, and must be read off by means of some 
form of reading microscope. 

Young's modulus may also be measured if the beam is 
fixed at one end and a weight is attached to the other end, the 
relation between Young's modulus and the deflection at the 
point of application of the force is in that case given by 

d = 4!W.g -rrn • 



SECTION XV. 

MODULUS OF RIGIDITY. 

Apparatus required: Uniform vnreSy cylindrical weight, 
and watch with seconds hand. 

The elastic reactions of a homogeneous body subject to 
strain depend on two constants, the bulk modulus, or resistance 
to change of volume, and the modulus of rigidit}^, or resistance 
to change of shape. 

If a cylinder AB oi radius r and length I has one of its ends 
A fixed, while the other B is twisted, the angle of twist 
(f) is connected with the moment of the twisting couple 
P, and the modulus of rigidity n by the relation 

Hence if P can be measured, n can be calculated 
from the dimensions of the cylinder. If the cylinder 
is thin, e.g. a wire, P may be determined by suspend- 
ing a body the moment of inertia of which is known 
from the wire, and letting it perform torsional oscilla- . 
tions. 

The time T of a torsional oscillation will be 



=V^' 



_ B 

Fig. 30. 



where I is the moment of inertia of the suspended body about 
the axis of twist, and N is the moment of the torsional couple 
per radian of twist, that is the value of P when </> = 1, i.e. 



XV 



MODULUS OF RIGIDITY 



87 



Hence 



r=27r 



and n = 



SttI 



11. 



j"i 



n 



If the suspended body is of some regular form its moment 
of inertia about the torsional axis can be calcu- 
lated from its mass and shape. Otherwise we 
may proceed as in the Exercise " Moments of 
Inertia II." p. 73, to determine /. 

In order to verify the truth of the law ex- 
pressed by the above equation, suspend a brass 
cylinder (Fig. 31) by means of three different 
wires of the same material but different dimen- 
sions, and determine the times of torsional oscil- 
lation (Section III.). In the example given below 
two of the wires are cut from the same coil, the 
length of one piece being about double that of 
the other, and the third is equal in length to the 
longer of the two former wires but is thicker. 
The diameters of the wires should be determined 
by the screw gauge at four different points and 
the means taken. 

Tabulate the results as follows : — 



Fig. 31. 



2 March, 1899. 
Steel wires marked C. 



Length 
/ 

cms. 


Mean 
raditur 

OOUI. 


r* 


I 


T 


r« 


^- 


83-5 

41-75 

83-5 


0165 
0165 
0295 


7-41xl0-» 
7-41 „ 
75-72 ,. 


8-87 X 10-" 
Ac. 
Ac. 


6-45 
4-54 
215 


41-60 

20-61 

4-62 


3-69 X 10-» 
Ac. 
<fec. 












Mean 





88 MECHANICS AND GENERAL PHYSICS XV 

Take the mean of the values obtained in the last column 
and substitute in the equation 

Stt/ 
I 

to determine the modulus of rigidity n. 

The value of / can be found from the weight and dimensions 
of the suspended body, since the moment of inertia of a cylinder 
about its axis is the product of its mass into half the square 
of its radius. 

Tabulate your results as follows : — 

2 March, 1899. 
Steel wires and cylinder marked C. 

Mass of cylinder ... ... ... 663 grams 

Radius of cylinder ... ... ... 2"27 cms. 

Moment of Inertia (calculated) = 1578 

.*. n for steel wire used = 8*5 x 10" dynes per sq. cm. 



SECTION XVI. 



VISCOSITY. 

Apparatus required : Two long capillary tribes, an inverted 
hell-jar y a small flask, microscope and stage micrometer. 

If two layei-s of a liquid at a small distance x- apart are 
moving with velocities Vj and v^, the faster moving layer tends 
to increase the velocity of the slower, the slower to decrease 
that of the faster. If S is the surface of contact of the two 
layers, and F the magnitude of the force exerted by one layer 
on the other, then 

X 

where rj is the " coefficient of viscosity " of the liquid. The 
object of the present exercise is to determine the value of t) for 
a given liquid. 

If a liquid flows along a capillary tube the liquid in contact 
with the wall of the tube is either at rest or only moves very 
slowly. We shall assume that it is at rest. The layer of liquid 
next to the one in contact with the walls moves with small 
velocity, the layer next to it with a greater, and so on, the 
liquid at the axis of the tube moving with the greatest velocity. 
For the same tube, the velocity of the layers will increase more 
rapidly from the walls to the axis for a liquid of small viscosity 
than for a liquid of great viscosity. The volume of the liquid 
which under given conditions flows through the tube will thei-e- 
fore depend on the viscosity of the liquid. 



90 



MECHANICS AND GENERAL PHYSICS 



XVI 



It may be shewn that the volume V is given by the 
equation 






where 



R = radius of the tube. 

p = excess of static pressure at the outlet of the tube over 

atmospheric pressure. 
rj = coefficient of viscosity of liquid. 
t = time in seconds. 

An inverted bell-jar is provided (Fig. 32), having a rubber 
stopper through which passes a bent tube, to the end of which 




Fig. 32. 



an inclined capillary tube of diameter 2R and length I is 
attached by a piece of rubber tubing. A strip of paper is 
gummed to the jar about half-way up, and a horizontal mark on 
it serves to indicate the level of the liquid in the jar. 

Place the lower end of the capillary tube about 10 cms. 



XVI VISCOSITY 91 

above the table, and under it put a beaker to catch any liquid 
issuing. Place a thermometer in the bell-jar. Gum a piece 
of paper to the neck of a 4 oz. flask, and determine the volume 
up to the paper by weighing the flask when empty and 
when full of water. Fill the bell-jar with water at the tem- 
perature of the room, to a level a little above the upper 
mark on the gum paper. The water will flow through the 
tube into the beaker. 

When the level of the water in the bell-jar has fallen to a 
convenient mark on the paper strip on the side of the jar, 
remove the beaker quickly and replace it by the flask, noting 
the time to a second. When the flask is filled to the level of 
the gum paper replace it by the beaker, notice again the time 
and mark on the paper strip the level of the water in the 
bell -jar. 

Determine the mean static pressure at the outlet of the 
tube by measuring the heights of the two marks on the bell-jar 
above the centre of the bore of the tube at the outlet, and 
taking the mean, or by measuring the heights of the two marks 
from the table and subtracting the height of the centre of the 
bore of the tube above the table. If // is the height, p the density 
of the liquid and g the gravitational acceleration, p = hgp. 

The diameter of the tube should be found by placing the 
tube vertical under the stage of a microscope having a scale in 
the eyepiece, in such a way that one end is in focus. The 
greatest and least diameters of the tube at one end are thus 
found in divisions of the eyepiece scale. Reverse the tube and 
find the diameter of the other end. If the four measurements 
are nearly equal take the mean, if they differ materially use 
another tube. 

To find the value of an eyepiece scale division in cms., 
remove the tube, place on the stage of the microscope a scale 
divided into tenths of millimeters, and find how many of the 
eyepiece divisions are equivalent to ten of the stage scale. 

Take a wider tube and determine by means of it the viscosity 
of the salt solution provided. Find the density of the solution 
by weighing the flask when filled with it up to the level of the 
paper. 



92 MECHANICS AND GENERAL PHYSICS XVI 

Tabulate your results as follows : — 

21 March, 1899. 

Microscope A. 

Divisions of eyepiece scale corresponding to 1 mm. = 61 '5. 

Value of a division = 00163 cm. 

Weight of flask filled with water =100 grams. 

Do. empty = 24 grams. 

Volume of flask =76 c.c. 

Weight of flask filled with solution =116 grams. 

.'. Density of solution = 1*21 

Viscosity of water. Tube A, 

Diameters (1)68-6, 68-4) ^o j- r 

(2) 67-7, 67 31= ^^^^^-"^"^^P^"^" 





= ]11 cm. 


Length 


= 65*5 cms. 


Vol. 


= 76 c.c. 


Pressure 


= 19-6 cms. water = 1 9230 dynes 




per sq. cm, 


Time 


= 450 sec. 


Temperature 


= 11^-5 C. 


7;at ir-oC. 


= 0103. 



And similarly for the observations with the solution. 



SECTION XVII. 



SURFACE TENSION. 



Apparatus required 

and stage micrometer. 



Glass tube and scale, microscope 



Heat in the blow-pipe tiame a piece of glass tubing, which 
has been thorouqhly cleaned irmde with water and dried, and 
when it is soft draw the two ends apart rapidly so as to form a 
fine capillary tube. 

Break off the capillary part of the tube, and repeat the 
drawing-out process on the rest of the tube till half-a-dozen 
capillary tubes have been obtained, two about 1, two about '7, 
and two about *3 mm. diameter. 

Break off from each tube about 
20 cms. of the most uniform por- 
tion. 

Mount vertically in a small 
clean glass vessel containing 
water a clean transparent gra- 
duated scale. Wet the scale a 
little and place a tube against 
it (Fig. 38) ; the tube will stick 
and will not need supporting. 
The water will ascend in the 
tube. When the water has 
reached its greatest height, raise 
the tube a few mms., so that the 
inner wall above the meniscus is 

wet, and read on the scale the levels of the meniscus and of 
the surface of the liquid in the vessel. 




Fig. 88. 



94 MECHANICS AND GENERAL PHYSICS XVII 

In taking the latter reading the level of the surface at a 
little distance from the scale should be 
read, since close to the scale the surface is 
curved upwards, thus: — Read A, not B. 

Gum a small piece of paper to the tube, 
2 mms. above the top of the column. ^ 



Raise the tube 5 mms., and again take 
readings. If the diflference of level of the 
meniscus inside the tube and of the liquid „. . 

outside is the same as it was previously, 
the tube is uniform. If there is much difference another 
tube should be substituted for it. 

Repeat the observations with two other tubes of different 
diameters, attaching a strip of paper to each. 

Now take another vessel, clean it thoroughly, fill it with 
20°/o ethyl alcohol, and take observations with the other three 
tubes, attaching strips of paper as before. Pour the alcohol 
back into the stock bottle. 

Tlie radii of the tubes at points 4"5 mms. below the strips 
of paper must now be found. 

To do this a microscope provided with a micrometer eye- 
piece is required. To determine in cms. the value of a division 
of the scale of this eyepiece, place on the stage of the micro- 
scope a scale graduated in tenths of a mm., notice how many 
divisions on this scale correspond to some convenient number 
on the eyepiece scale, and hence find the value of a division of 
the eyepiece scale in centimetres. Remove the stage micro- 
meter, break off one of the tubes 45 mms. below the gum 
paper by marking it with a file or rough-edged knife, and 
mount each part with the broken section upwards on a micro- 
scope slide, attaching them by a little soft wax. Now place 
one of these mounted tubes on the stage, focus the broken 
end of the tube, and read its diameter in scale divisions 
of the eyepiece micrometer. If it is not accurately cir- 
cular, measure its greatest and least diameters and take 
the mean. 

Repeat with the other piece. Take the mean and convert 
to cms. 



XVII 



SURFACE TENSION 



95 



Do this with each capillary tube used, and calculate the 
Talue of the surface tension T in dynes per cm. from the 
equation 

^ "" 2 ' 

where 

g = value of gravitational acceleration. 

p = density of liquid. 

h = height of meniscus in tube above liquid outside. 

r = radius of tube. 

Tabulate your results as follows : — 

12 February, 1899. 

Microscope A. 

10 divisions of eyepiece scale = 1*67 divisions on stage scale 

= 167 mm. 
.'. 1 division of eyepiece scale = 00167 cm. 



Liquid 


Tube 


Reading in cms. at 


Height 
cm. 


Badius 


h.r. 


T 


Meniscus 


Base 


eyepiece 
divisions 


cms. 


Water 


■1 

3 


3-66 
3-65 
6-65 
6-64 


1-52 
1-51 
1-53 
1-52 


2141 
2-14/ 
512\ 
512| 


42-5 


•071 


•152 


742 


Alcohol 
20 V, 


4 
5 
6 

















Verify the value found for water by the following method : — 
Clean carefully the thin sheet of glass provided, and sus- 
pend it by means of the frame provided from the left-hand 
beam of a simple balance. Adjust it till its under edge is hori- 
zontal, and place weights in the right-hand pan till the pointer 



96 MECHANICS AND GENERAL PHYSICS XVII 

of the balance reads zero. Fill a clean glass vessel, wider than 
the strip of glass, with clean water, place the vessel under the 
strip and raise it gradually till the lower edge of the glass 
just touches the surface of the water. Push the glass a 
millimetre down into the water so as to wet the edge, and then 
place weights in the right-hand pan of the balance till the edge 
is just torn away from the water. If m grams, in addition to 
the weight to balance the glass in air, have to be used, and if I 
is the length of the edge of glass touching the surface, 

7ng=:T.2l, or T^^""^^^. 

Record as follows : — 

Weight to balance glass in air = '52 gram 

„ „ „ touching water = 1'59 granivS 

Difference due to surface tension = 1*07 „ 

Length of edge touching water = 7'5 cms. 

Surface tension of water =70 dynes per cm. 



BOOK III. 

HEAT. 



SECTION XVIII. 

COEFFICIENT OF EXPANSION OF A SOLID. 

Apparatus required : Tubes of metal and glass, flow of 
hot water or steam, thermometer and reading microscope. 

When the temperature of a solid is raised, the solid gene- 
rally expands, and if the distance between two given points in 
the solid is l^ at 0° C. and U at f C. we have, to a sufficient 
degree of approximation, 

lt = Ui\-Vat), 

where a is a constant called the coefficient of linear expansion 
or dilatation. 

In order to determine this coefficient, a bar of convenient 
length may be measured at two temperatures ^ and t,. If 
^ and /a are the observed lengths, we have as a consequence of 
the above relation 

As a is small in the case of solids, we may as a rule with 
sufficient accuracy substitute the length at the ordinary tem- 
perature of the room for i,. Thus the coefficient of expansion 
of metals generally lies between 00001 and "00004, and their 
linear dimensions will increase therefore by less than a 
thousandth part of the original length between 0* and 20** 
If therefore it is not desired to measure a to an accuracy 
greater than one part in a thousand, l^ may be taken as 
identical with the length measured at the ordinary tempera- 
8. p. 7 



98 



HEAT 



XVIII 



ture of the room, and with rods about half a metre long an 
ordinary scale graduated to millimetres will be sufficient for 
the purpose of measurement. 

The quantity Za — h being small some method of measure- 
ment suitable for small lengths must be used, e.g. a microscope 
provided with a graduated scale in the eyepiece. 

The heating of the material to be tested is carried out very 
conveniently if it is given in the form of a hollow rod, since hot 
water or steam may be sent through it, to raise it to the 
required temperature. 

Two tubes of about 60 cms. length are provided, one of glass 
and one of brass ; and are mounted so that steam or hot water 
may be passed through them. If two reading microscopes are 
available, two marks may be made on the tubes one near each 
end, and the displacements of these marks measured after a 
temperature change which should not be less than 70° C. The 
apparatus provided gives sufficiently accurate results with one 
reading microscope only. 

A notch' is cut near one end of the tube, and this notch with 
the tube placed in a horizontal position is made to rest on the 
knife edge of the stand provided. For additional security an 
arrangement is made by means of which the rod may be 
clamped above the knife edge. The other end of the rod may 
rest on the microscope stage, but in the stand provided it is 
supported by passing loosely through a hole cut into a piece of 
wood fixed to the stand. 




Fig. 35. 

Clamp the stand firmly to the bench or slab on which the 
experiment is to be performed. The free end of the tube 



XVIII 



COEFFICIENT OF EXPANSION OF A SOLID 



99 



carries a narrow paper label with a fine pencil line, drawn at 
right angles to the tube. 

You are provided with a microscope having an objective of 
low power so that it is capable of including a field of view of 
about 2 mm. diameter. The eyepiece of the microscope carries 
a micrometer scale. Place the microscopes in such a position 
that the pencil line is near the centre of the field and then 
clamp the microscope firmly to the table or slab (Fig. 35). 

Attach rubber tubes to the ends of the experimental tube 
and send a stream of cold water through ; observing the tem- 
perature of the water by means of a thermometer placed in the 
issuing stream. Observe the reading in the microscope, turn- 
ing the scale until its divisions are parallel to the pencil line. 

Send steam or hot water through the tube and again take 
readings. Finally repeat the experiment with cold water. 

Substitute the second tube and test similarly. Measure 
the length between the notch and the pencil line to the 
nearest mm. 

Determine the value of a scale division of the eyepiece by 
placing a scale graduated in -J^ mm. on the stage of the 
microscope section. 

Record and calculate the coefficient as follows : 
Date, June 8, 1899. 
Microscope A. 
58 divisions of eyepiece scale = 10 on stage scale 

= one millimetre 
.'.1 division of the eyepiece = 00172 cms. 

Brass tube 58*6 cms. between marks. 



Temp. 


Reading 


Difference 


Scale 


Cms. 


Water 30' C. 
Steam 100 
Water 30 


1-42 
5-72 
1-42 


4-30 


•0744 



•0744 
• • ^ 68-6 X 70 
Similarly for the glass tube. 



•0000181. 



7—2 



SECTION XIX. 

THERMAL EXPANSION OF A LIQUID. 

Apparatus required : Bulb with graduated stem, and ice. 

The change of volume of a liquid with increase of tempera- 
ture is most conveniently found by enclosing the liquid in a 
vessel made of a material of which the coefficient of expansion 
is known, and observing the apparent change of volume of the 
liquid as the temperature of liquid and containing vessel is 
varied. 

Since the apparent expansion is, in general, only a small 
fraction of the total volume, the volume of that part 
of the vessel in which the expansion is measured ^ 
should only be a small fraction of the total volume. o 

The vessel, which is called a " dilatometer," is 
generally a glass bulb provided with a graduated stem 
having a fine bore, the liquid filling the bulb and part 
of the stem (Fig. 36). 

If at 0° C. the surface of the liquid in the stem 
stands at the division Uo of the stem, and the volume 
of the bulb up to the zero division of the scale is m, 
measured in scale divisions, the volume of the liquid 
at 0'' C. is m + tiq scale divisions. 

If when the dilatometer and liquid are at f C. the 

surface ,of the liquid reads n on the scale, we have, for 

the contents of the vessel up to the mark n at tem- „. 

^ Fig. 36. 

perature f C, {m + n){l + at), where a is the coefficient 

of expansion of the vessel. If further fft is the expansion of unit 



XIX THERMAL EXPANSION OF A LIQUID 101 

volume of the liquid between 0° and i°, the volume of the liquid 
at the latter temperature will be (m + n©) (1 + I3t), hence : 
(m + n) (1 + at) = (m + Uo) (1 + A) 

orl+A = -^^(l+«0 
'^ m + 7?o 

= 1 + ^ -\- at approximately. 

m 

from which the expansion /8f may readily be determined. 

Unit volume at the freezing point of water will occupy a 
volume 1 + ^^ at temperature t. 

A graduated capillary tube is provided, at the end of which 
a bulb of about 1 cm. diameter has been blown. 

Wash and dry the bulb and tube and then weigh. 

Heat the bulb gently in a bunsen burner, then dip the open 
end of the tube into a small quantity of the liquid to be tested 
which, if necessary, has been boiled to free it from dissolved air. 
As the air in the bulb contracts, liquid is drawn up the tube 
into the bulb. Repeat the heating and cooling till the bulb 
and a short length of the stem are filled with liquid. 

Place the tube in a water bath at about 16° C, and after 
waiting 10 minutes to allow the liquid in the bulb to take up 
the temperature of the bath, read the temperature of the bath 
and the position of the liquid in the stem. 

Weigh the bulb and contents. The difference between this 
weight and the previous one is the weight of the liquid in the 
bulb, and if its density is known, the volume of the liquid, 
therefore the volume of the bulb and stem containing it, can 
be found. 

To determine the volume of one scale division of the stem, 
heat the bulb gently till the liquid stands rather more than 
half way up the stem, then plunge the open end of the stem 
into clean mercury, which will be drawn up the tube as the 
liquids contract. Allow the dilatometer to cool till the posi- 
tions of both ends of the mercury thread can be read on the 
scale. 



102 



HEAT 



XIX 



Observe the length of the thread and weigh the bulb and 
its contents again. The difference between this weight and 
the last one is the weight of mercury, which may be divided 
by 13'.596 to give the volume, and hence the volume of a scale 
division in c.c. may be found. 

The dilatometer should again be heated till the mercury 
thread is expelled from the tube, and the upper end of the 
tube then sealed in the blowpipe (Fig. 36). A label is attached 
to the tube giving the mass of liquid in the tube and the value 
of a scale division, from these numbers the quantity m may be 
calculated. 

If the liquid is water particular care must be taken with 
the observations between 0° and 10° owing to the anomalous 
behaviour of that liquid. The dilatometer should first be 
placed in a bath containing melting ice, and after the position 
of the liquid has become constant, a reading should be taken. 
The temperature of the bath should then be raised about 
two degrees, and observation of temperature and apparent 
volume taken at the new temperature. The observation 
should be repeated at intervals of 2° C. up to 10° C. and 
then at intervals of 5° C. over the range of temperature desired, 
and again as the temperature is decreased. 

The observation and results should be tabulated as follows : 

Expansion of Water. 

10 February, 1899. 

Dilatometer A, m = 15500, a = '000020. 



Temp. 


Reading 


n-iiQ 


n — iiQ 
m 


at 


^t 


0°C. 


24-0 














0-6 


23-2 


- -8 


- 0000516 


•000012 


- -000040 


1-7 


22-0 


-2-0 


-•000129 


-000034 


- -000095 


3-3 


20-4 


-3-6 


- -000232 


•000066 


- -000166 


5-3 


2M 


-2-9 


- -000187 


•000106 


- -000081 


7-3 


22-2 


-1-8 


-•000116 


•000146 


+ -000030 


12-3 


28-0 


+ 4-0 


+ -000258 


•000246 


+ ^000504 


&c. 


&c. 




&c. 







XIX 



THERMAL EXPANSION OF A LIQUID 



103 



Numerical calculation will be facilitated, if the reciprocal of 
m is first calculated and then multiplied successively by the 
numbers in the thiid column. Three figures are sufficient and 
Crelle's Tables may be used. 

A curve (Fig. 37) should be drawn with the observed 
temperatures as abscissae and total volumes 1 + y9< as ordiuates. 





— 






r 


















" 






































































1003 






































































































































































































































.' 










l-OOl 




































y 














































'' 








- 




































„* 






































*' 












































-• 


























1000 
















t 


'' 
































'- 


'♦<. 




^ 


_,, 


i' 


' 























































































































































































































































1 


0' 


















2 


0' 


_ 





Fig. 37. 

The numbers, giving the change of volume of water, should 
be taken out of some book of physical constants, and repre- 
sented by a second curve for comparison. 

Care must be taken to wait sufficiently long after each 
change of temperature to allow the water in the bulb to come 
to its final state, for it must be remembered that the thermal 
conductivity of water is small, and that convection currents are 
diminished in efficacy owing to the small size of the bulb. The 
numbers taken with rising and falling thermometer should be 
plotted separately and should shew no systematic difference. 



SECTION XX. 

COEFFICIENT OF INCREASE OF PRESSURE OF A 
GAS WITH TEMPERATURE. 



Apparatus required : Bulb and pressure apparatus, water 
bath, thermometer. 

If the temperature of a constant volume of gas is raised, 
the pressure of the gas varies according to the equation 
^j = p^ (1 + ^t), where po is the pressure at 0° C, t the tem- 
perature Centigrade, and /S a constant, called the coefficient of 
increase of pressure with temperature. 




Fig. 38. 



XX INCREASE OF PRESSURE WITH TEMPERATURE IN AIR 105 

To verify the law expressed by the above equation in the 
case of air the apparatus shewn in Fig. 38 is provided. 

A is SL glass bulb to which a tube of small bore BC is 
attached, i) is a similar tube. The lower ends of the tubes 
are joined together by a piece of rubber tubing which can be 
compressed by turning a screw which acts on the plate E. 

The rubber tube and the lower part of the tubes C, D, are 
filled with mercury. 

By adjusting the quantity of mercury and the pressure at E 
the surface of the mercury in the left-hand tube may be brought 
to the fixed mark C, If the position of the surface of the 
mercury in D is then read, and the height of the barometer is 
known, we have the total pressure to which the air in A is 
subjected. 

To determine how this pressure varies with the temperature 
of the air in the bulb, fill the bath in which the bulb is 
placed with cold water, noting its temperature. Adjust the 
mercury to C, and read the positions of D on the scale along- 
side. 

Raise the temperature of the bath about 5° C. keeping the 
water stirred, and repeat the adjustment and readings. Take 
four observations at intervals of 5° C. and then wait until the 
thermometer has risen 25°, i.e. stands 40° higher than at the 
beginning of the observations. Take four more observations at 
intervals of 5°. Then release the pressure of the screw so as to 
lower the mercury in the tube C, and pour sufficient cold water 
into the bath to lower its temperature 5° C. Stir well and 
raise the mercury to C again, then read D. Lower the mercury, 
cool further and repeat the observations till the original tem- 
perature is reached. Take the means of each pair of observations 
at nearly the same temperature and use them in what follows. 

Read the barometer ; the pressure of the gas in terms of a 
column of mercury will be the sum of the height of the 
barometer and the diflference in level between C and D. 

If p is this pressure and v the volume of the gas at 
temperature t, we have pv = p^Voil ■¥ ^t), where Vo is the 
volume at 0°C. ; t; will increase slightly as the temperature 
rises owing to the expansion of the glass. 



106 HEAT XX 

If the coefficient of expansion of the glass is a, the volume 
of the vessel v is obtained from v = Vo(l + at). 

Hence P = ^«H2- 

Or since at is, in the above experiments, always less than 
002, we may write 

p=Po(l-\-{^--oi)t). 

If piti, pit^ are two sets of readings of p and t, we may form 
two similar equations and eliminating ^o find 

Pi ^ Pi 

(l+(/3-a)^,) H-(/3-a)^' 

and hence 




p^ Pi 

/8 may therefore be calculated, if a is known, by observation 
of any two corresponding pressures and temperatures. If a 
number of readings are taken at regular intervals of tempera- 
ture we may apply the method explained in Section (III) and 
secure the smallest probable error by taking a certain number 
of observations, omitting the same number, and finally taking 
again an equal number. The observations are combined and 
reduced as given in the Table below. Of the two forms for 
(/S — a) given above the first involves less arithmetical labour, 
and would naturally be taken in the case that the pressure is 
observed only for two temperatures. But for a series of 
experiments the results may be tabulated in a more sym- 
metrical manner if the second form is adopted. The labour of 
calculation is reduced if the reciprocals of pi and jh are 
obtained from Barlow's Tables. 

The fraction — etc. as well as the final quotient may be 
Pi 

found with sufficient accuracy from Crelle's multiplication 
tables. 



XX INCREASE OF PRESSURE WITH TEMPERATURE IN AIR 107 



Arrange the observations as follows : 

Apparatus marked K. 

8 Nov., 1899. 

Height of Barometer 76*2 cms. 







1 


'i 




1 


h 


1 1 


t- t^ 




h. 


Px 






h I'a 


— 








/3-0 






Pi 


l>i 




P% 


Pi 


Pi Pt 


Pi Pi 




10°-5 


77-7 


•01287 


•135 


60-3 88-4 


•01131 


•567 


•00156 


•432 


•00361 


15-2 


78-9 


1267 


•191 


55-1 89-6 


1116 


•609 


151 


•418 


361 


20 


801 


1248 


•249 


60 90-9 


1100 


•660 


148 


•411 1 


360 


26-0 


81-3 


1230 


•307 


65-0 92-1 


1086 


•706 1 


144 


•399 


361 



Mean value of (y9 - a) 
For glass of bulb a . . . 
yS for air 
1 
fi 



= -003608 
= 00002 8 
= 003636 

= 275° 



According to Regnault fi should be equal to '0003669. 
There are two sources of error both tending to reduce the 
apparent value of fi. One is the expansion of the glass vessel 
due to increased internal pressure at the higher temperatures, 
and the other is due to the fact that the air in the tube leading 
from the bulb is not at the same temperature as that in the 
bulb. The first named error is not likely to be more than 
•000003 with the dimension of apparatus used, the second error 
depends on the width of the bore, which should therefore be 
narrow. 

If the bulb is not properly dried, too high value will be 
obtained owing to the evaporation of the water from the walls 
of the vessel. 

The increase of pressure of air is used to measure tempera- 
tures in the so-called "air thermometers," the construction of 
which is the same in principle as that of the apparatus used in 
this exercise. 



SECTION XXI. 

COEFFICIENT OF EXPANSION OF AIR AT 
CONSTANT PRESSURE. 

Apparatus required : Graduated tube and index, water 
bath, and thermometer. 

When the temperature of a gas maintained at constant 
pressure is raised, the volume increases, and may be repre- 
sented by an expression of the form 

v = Vo(l+at), 

where v is the volume at ^°C., Vo at 0°C. and a is a constant 
called the coefficient of expansion of the gas. 

If the gas is enclosed in a vessel the coefficient of cubical 
expansion of the material of which is /3, the apparent volume 
of the gas will be expressed by the equation 

v = Vo(l+a^l3t). 

If ti and t^ are two temperatures and Vi, v^, the corresponding 
apparent volumes, it is found eliminating Vq that 

1_1 

_ _ 

Hence if observations of the variation of volume of a given 
mass of gas, enclosed in a vessel of known coefficient of expan- 
sion, are made, the coefficient of expansion of the gas may be 
determined. 

The apparatus provided consists of a graduated glass tube 
of fine bore, one end of which is sealed, and the other attached 



XXI 



EXPANSION OF AIR AT CONSTANT PRESSURE 



109 



to a wide tube bent at right angles. The complete tube is 
attached to a brass frame. The capillary tube is filled for 
about two-thirds of its length with air, which is confined by a 
thread of strong sulphuric acid. This acid thread also serves to 
indicate the volume of the air and to keep it dry (Fig. 39). 
The wider part of the tube serves for introducing the acid, so 
as to enclose a convenient quantity of air. 





Fig. 39. 



Fig. 40. 



To find the eflfect of change of temperature on the enclosed 
air, place the graduated tube horizontal in a water bath the 
temperature of which can be varied (Fig. 40). Mix ice and if 
necessary salt with the water till the temperature indicated on 
a thermometer placed in the water is 0°C. Stir the water well 
and aft^r waiting till the thread of acid has taken a constant 
position, take a reading of that end of it which is in contact 
with the gas. Increase the temperature of the water by 5° C, 
and again observe the volume. Take four observations increas- 
ing the temperature in steps of 5^ C, then wait until the tem- 
perature has risen another 25° and take four more observations 
at temperatures increasing by 5°C. 

Then decrease the temperature and take observations at 
approximately the same temperatures as during the rise. 

The closed end will not in general be at the zero of the 



110 



HEAT 



XXI 



scale and its reading must therefore be subtracted from the 
readings of the index to obtain the volumes. As the closed 
end is often conical a short thread of acid may be introduced 
to occupy the conical spaces. It is assumed that the bore is 
uniform so that the differences in the readings of the scale 
are proportional to the volumes. 

The observations are reduced as in the previous section. 

Enter as follows : 

Apparatus marked B. 

12 November, 1899. 

Reading at closed end 5*05. 



Temperature increasing 


Temperature decreasing 


t 


Beading 


Vol. 


t 


Eeading 


Vol. 


0-0 

5 

10-1 

15-1 


15-25 
15-42 
15-59 
15-80 


10-20 
10-37 
10-54 
10-75^ 


•0 

4-8 

10-2 

15-1 


15-25 
15-42 
15-61 
15-76 


10-20 
10-37 
10-56 
10-70 


39-5 
45-0 
49-8 
54-6 


16-67 
16-83 
17-03 
17-23 


11-62 
11-78 
11-98 
12-18 


40-5 
45-2 
50-6 
55-6 


16-71 
16-87 
17-07 
17-27 


11-66 
11-82 
12-02 
12-22 



Take the mean of the numbers for increasing and decreasing 
temperatures and enter as in the following table. 







1 


h 






1 


h 


1 1 


ig «l 




tl 


^1 


— 




«2 


^2 


— 









a-/3 




^'i 


^1 






^2 


^2 


^^1 ^2 


V2 Vi 




o°c. 


10-20 


-0980 





40° -0 


11-64 


-0859 


3-436 


-0121 


3-436 


-00353 


4-9 


10-37 


•0964 


•472 


45-1 


11-80 


-0847 


3-819 


-0117 


3-347 


•00349 


10-15 


10-55 


-0948 


-962 


50-2 


12 00 


-0833 


4-182 


-0115 


3-220 


•00357 


15-1 


10-72 


-0933 


1-409 


55-1 


12-20 


•0820 


4-520 


-0113 


3-111 


•00363 



Mean value of a — yS 
^ for glass used 



.*. a 



= -00355 
= -00003 



= -00358 



XXI 



EXPANSION OF AIR AT CONSTANT PRESSURE 



111 



For gases satisfying Boyle's law the coefficient of expansion 
at constant pressure is the same as that of increase of pressure 
for constant volume. The value of a found is about two per 
cent, too small. 

Take temperatures as abscissae, and volumes in scale divi- 
sions as ordinates and draw a straight line passing as nearly as 
possible through the points so determined (Fig. 41). From the 
inclination of the straight line find the value of a — yS. 























































































































|0,ri_ 


















































, 






















































^ 


^ 


\^ 


















































f^ 


^ 














v..) 




































^ 


-^ 


















































^ 


^ 


'^ 
























1 1 rk .. 




























'^ 
















1 


































^ 


^ 




















































^ 


















































<^ 




















































w^ 


^ 


■^ 





























































































































































































































































































10 



20 T>4Dp 30 
Fig. 41. 



40 



60 



SECTION XXII. 

EFFECT OF PRESSURE ON THE BOILING POINT 
OF A LIQUID. 



Apparatus required : Thermometer, flask, condenser and 
air-pumps. 

When the pressure under which a liquid boils is increased, 
the temperature to which the liquid must be brought before 
boiling takes place is raised, and if the pressure is decreased 
the temperature at which boiling occurs is reduced. It is the 
object of the present exercise to find the relation between 
the two changes. 

In the diagi-am of the apparatus to be used (Fig. 42), ul is a 
Bunsen filter pump through which a rapid flow of water from 
the tap T^ may be kept up. 




Fig. 42. 



XXII RELATION BETWEEN PRESSURE AND BOILING POINT 113 

(7 is a mercury manometer, with a reservoir R for adjusting 
the reading of the surface in the open tube to zero of the scale 
alongside. 

i5 is a trap arranged so as to prevent the water from the pump 
flowing back to the manometer when the pump is stopped. 

i) is a three-way cock having a central passage parallel to 
the handle, this passage joins at the centre to another passage 
at right angles leading only to one side. The side to which 
it leads is marked by a black dot. When the handle is vertical 
with the black dot to the right, the pump is in connection 
Mrith the condensing apparatus. Call this position A. When 
the black dot is to the left the apparatus is open to the air. 
Call this position B. When the handle is horizontal with the 
black dot below, the way to the bottle is closed, and the pump 
is in connection with the air. (Position C.) 

J5^ is a stop-cock to cut off the boiling apparatus from the pump. 

^ is a bottle to diminish the changes of pressure due to the 
bursting of bubbles in the flask N during boiling. 

r is a thermometer giving the temperature of the steam 
rising from the liquid experimented on — water in this case. 

J/ is a condenser in which the steam is condensed, the 
condensed liquid running back into the flask. 

To commence the experiment, put the flask N in connection 
with the air by turning D till the handle is in the position B. 
Set the water in the flask boiling, and see that water is passing 
slowly through the condenser from the tap T^. Read the 
barometer, and calculate the true boiling point of water at the 
observed pressure, taking the boiling point as changing '37° C. 
per cm. change of pressure. When the temperature is steady, 
read the thermometer T, and determine the error at the boiling 
point. The whole of the mercury thread should be inside the 
flask. If a considerable portion of it lies outside a correction 
called that of " emergent stem " becomes necessary. 

If the error of the thermometer at zero has been previously 
determined and marked on the thermometer, the correction at 
any reading may be found with sufficient accuracy for the 
present purpose by drawing the curve of corrections as in the 
Intermediate Course of Practical Physics, 

8. p. 8 



114 



HEAT 



XXII 



Now turn D into position A, and set the pump working. 
When the difference of level of tbe mercury columns in G is 
4 cms., turn the stop-cock into position C, turn off the water at 
tap Tj, and, after waiting two minutes, read the thermometer 
and manometer three times at half-minute intervals. After 
taking the observations, turn on the water again, place D 
vertical with the dot to the right, and diminish the pressure 
4 cms. further. Take readings as before. 

Repeat the observations at intervals of 4 cms. pressure till 
the temperature is about 75"; then with the pressure increasing 
again to that of the atmosphere. Take the mean of each set of 
three, correct it for the errors of the thermometer, and tabulate 
your results as follows : — 

1 April, 1899. 

Observed height of barometer ... 

Calculated boiling point 

Observed reading of thermometer 
atmospheric pressure 

Hence correction of thermometer at boil- 
ing point 



... 


763-2 mms. 


... 


100°-1 C. 


at 




vil 


99-3° 


)ll- 


= -h-8° 



Pressure 


Temperature 


at 
manometer 


in 
flask 


Observed 


Correction 


Corrected 




40-2 

78-1 

121-2 

162-1 

&c. 


763-2 
723-0 
685-1 
642-0 
601-1 


99-3 

97-8 
96-3 
94-6 
92-8 


+ •8 
•8 
-8 
•8 
•8 


100-1 
98-6 
97-1 
95-4 
93-6 
<fec. 



Now plot a curve with pressures as abscissae, and the 
corrected temperatures as ordinates, representing the points 
obtained with the pressure decreasing by a cross, those with 
the pressure increasing by a circle. 

The two sets of observations should not differ by more than 
about '1° from each other. If they do, sufficient time has not 



XXII RELATION BETWEEN PRESSURE AND BOILING POINT 115 

beeD allowed between each observation for the temperature of 
the steam to become constant. 

From the observations calculate the temperatures corres- 
ponding to pressures of 760, 720, 680 mms., etc., by graphical 
construction or as follows : — 

Let pi be the observed pressure which is nearest to the 
pressure p for which the temperature t of boiling is to be 
determined. Let p^ be a second observed pressure near p. 
Then if ^ and ^2 be the temperatures of boiling corresponding 
topi.p^ 

Pi p\ 

The second term will be less than one degree, if the 
observations have been taken according to the above in- 
structions. Hence this term need only be calculated to one 
significant figure. 

Tabulate the results along with Regnault's as follows : — 



Pressure 


Temperature 








Experiment 


Begnault 


760 mms. 


100-0 


100° OC. 


720 


98-6 


98-5 


680 




96-9 


640 




95-2 


600 




93-5 


560 




91-7 


520 




89-7 


480 




87-6 


440 




85-4 


400 




830 


360 




80-4 


320 




77-5 



8—2 



SECTION XXIII. 



HYGROMETRY. 



Apparatus required 

bulb thermometers, ether. 



DanielVs hygrometer, wet and dry 



The most important determination in connection with the 
aqueous vapour present in the atmosphere is that of the 
"relative humidity" or "fractional saturation," i.e. the ratio of 
the amount of moisture actually present at any time in a given 
space, to the amount which would at that temperature saturate 
the space. This quantity may be determined directly by 
passing a known volume of moist air through tubes filled with 
drying materials and finding the increase in weight of these 
materials due to the moisture absorbed. It is determined in 
practice however by one of two methods : — dew point observa- 
tions, or wet and dry bulb observations. 

In the first of these methods the air is cooled down till the 
vapour present in it begins to condense, and 
the temperature at which condensation begins 
is observed. The instrument used is called a 
dew point hygrometer. The form known as 
Daniell's hygrometer consists of a tube with a 
bulb at each end containing ether and vapour 
of ether only. The outer surface of the lower 
bulb is silvered, gilded or made of black glass 
and encloses a thermometer ; the upper bulb is 
covered with muslin (Fig. 43). A thermometer 
fixed to the stem indicates the temperature of 
the air. 

Determine the dew point at a quiet spot outside the 
laboratory, in the following way. Tilt the instrument till the 




Fig. 43. 



XXIII HYGROMETRY 117 

ether has run into the lower bulb. Pour a little ether on the 
muslin surrounding the upper bulb. As this ether evaporates 
the bulb is cooled and the ether in its interior condensed. The 
ether in the lower bulb now evaporates and is cooled. Watch 
carefully the polished surface of the lower bulb to see when 
moisture is first deposited on it. Keep the muslin saturated 
with ether, and occasionally shake the instrument so as to 
stir the liquid in the lower bulb. Note the temperature of 
the air and of the thermometer in the bulb, when moisture 
first appears. The instant of appearance may be best detected 
by watching the image of some object in the polished surface 
of the bulb. Cease appl3dng ether to the muslin and watch 
when the last sign of moisture disappears. Note again the 
temperatures and take the mean of the readings in each case. 
The mean temperature indicated by the thermometer in the 
bulb is the dew point. Find from tables the pressure of 
aqueous vapour at the dew point, and at the temperature of 
the air. The quotient is the relative humidity, or fractional 
saturation. 

Record as follows: — 

16 March, 1897. 

Temp, of air = 17*0° C, .*. pressure of vapour = 1*44 cms. 

of mercury 
„ dew point = 12*5° „ „ =1*08 cms. 

of mercury 

Relative humidity = — j^; = '*^^- 

The second method consists in observing the difference of 
readings of two thermometers, one with its bulb surrounded by 
a wet wick and the other with its bulb free. The arrangement 
is known as the wet and dry bulb thermometer (Fig. 44). 

The indications of the wet bulb do not depend, however, 
exclusively on the hygrometric state of the air, for if the air 
near the t hermometers is in motion, evaporation is more rapid 
and the reading lowered in consequence. The readings can 



118 



HEAT 



XXIII 



only be interpreted by means of tables based on experiments, 
but it is found by comparison with more accurate 
hygrometers, that if the air round the thermo- 
meters is neither absolutely still nor violently 
disturbed, these tables give results which are 
sufficiently accurate for meteorological purposes. 

Saturate the muslin surrounding the wet bulb 
of the hygrometer provided, and the wick hanging 
from it, with water, and place it in a position so 
that the air can flow across the bulbs of the 
thermometers. If the air is still, suspend the 
instrument by a string from a fixed point, and 
set it slowing swinging as a pendulum. 

The temperature of the wet bulb will be found 
to decrease. When it becomes steady, read both 
the thermometers, and use table 35, Lupton, p. 30, 
to find the pressure of the aqueous vapour present 
in the air. Then divide by the pressure of 
saturated vapour at the temperature of the dry bulb. The 
quotient is the relative humidity, or fractional saturation. 

Record as follows : 

Temp, of air = 17° C, .*. pressure of vapour = 1*44 cms. 

„ wet bulb = 14-5° „ „ „ =1-09 „ 

109 
Relative humidity = r-^^T = *'^^- 




Fig. 44. 



SECTION XXIV. 

LAWS OF COOLING. 

Apparatus required: Calommeter, with (a) simple en- 
closure, (b) enclosure luith water jacket ; glass tubing to be used 
as pipette; thermoineter graduated in -^ degrees and reading 
frorti about 10° to 25° C; thermometer graduated in -^ degrees 
and reading up to 50° C. 

If a body differs in temperature from its surroundings, it 
will lose or gain heat by radiation and by conduction to the 
bodies surrounding it. 

The conduction is much affected by air currents caused 
by the air in contact with the calorimeter ascending and 
being replaced by colder air. Conduction is thus increased 
by the fresh supply of cold air, and is rendered irregular if 
by any cause the movement of the air is accelerated or 
retarded. Heat earned away by the bodily motion of the 
air is said to be carried by "convection." 

Experiment I. To determine the rate of cooling of a 
calorimeter under various conditions. 

Draw a pencil line round the inside of the calorimeter 
(Fig. 45) provided, about 1 cm. below the rim. 

Fill it to the mark with water at a temperature about 20° 
above that of the room. Place the calorimeter on the table 
unprotected, and place a thermometer, reading to 50°, in the 
water. Take readings of the temperature of the water every 
half minute during a period of two minutes, stirring the 
water well the whole time, and, estimating to 0^01 C, wait 
two minutes and then take live further observations at half 
minute intervals. Take the temperature of the air by means 
of the smaller tliermoraeter in the interval of waiting. 



120 



HEAT 



XXIV 



Kaise the water to its original temperature, and repeat the 
experiments with the calorimeter protected by an outside vessel 
but not covered ; and thirdly, with the calorimeter covered. 




CI=3> 



Fig. 45. 

The experiments are entered and reduced as follows : 
12 Oct. 1898. 
1. Calorimeter unprotected. 



Time 
hr. min. 


Temperature 


Time 
hr. min. 


Temperature 


Fall in 4 min. 


3 10-0 
10-5 
110 
11-5 
12^0 


38°^18C. 

•10 

•00 
37-94 

•86 


3 140 
14^5 
150 
15^5 
16-0 


37°-53C. 

•46 
•37 
•29 
•23 


•65 

•64 
•63 
•65 
•63 


Means 


38^016 




37-376 


•640 


Mean temp. 37° •696 









Average fall in four minutes 

„ per minute 

Temperature of air 



•640 

•160 

18°-5 C. 



XXIV 



LAWS OF COOLING 



121 



'ti 



The rates of cooling when the calorimeter is protected by 
an outside vessel, and covered or uncovered, must be reduced 
in exactly the same way. Compare the results in the three 
cases, and draw your conclusions. 

Experiment II. Having seen how the effects of con- 
vection currents can be diminished by properly protecting the 
calorimeter, we must next study how the amount of the cooling, 
in nearly quiescent air, varies with the temperatures of the 
body cooled and of its surroundings. 

You are provided with a calorimeter which can be placed 
in an enclosure consisting of a vessel having 
a double wall, into which water at the 
temperature of the room may be poured 
(Fig. 46). Two thermometers graduated 
in tenths give the temperatures of the water 
in the calorimeter and enclosure respec- 
tively. 

Fill the calorimeter with water up to 
the pencil line drawn round the inside 1 cm. 
below the rim. Raise the water to a tem- 
perature approximately 5° above that of the 
room, and observe the temperature every 
half minute for two minutes, then wait two 
minutes and observe again for two minutes, 
keeping the water well stirred. Note the 
temperature of the enclosure, which should 
be approximately the same as that of the 
room, at the beginning and end of the 
observations. 

Tabulate as shown previously. 

Raise the water in the calorimeter to temperatures 10°, 20°, 
and 30° above that of the room, and observe the rate of cooling 
in each case, adjusting the temperature of the enclosure to be 
the same within a few tenths of a degree during each set of 
observations. 

Calculate in each case the ratio of the rate of cooling to the 
excess of the mean temperature of the calorimeter over that of 
the enclosure. Call that ratio k. 



I. 



XZZ3Z 



Fig. 46. 



122 HEAT XXIV 

Collect your results as follows : 

Rate of Cooling per min. 
Mean temperature Mean temperature Cooling per per degree excess 

of calorimeter of enclosure Difference minute =h 

46-54° C. 14-30° C. 32-24° C. -494 -0153 

37-31 14-25 23-06 -281 0122 

27-58 14-45 13-13 -158 -0120 

19-66 14-40 5-22 -056 -0107 

Mean -0126 

The quantity called h is the rate of cooling for a difference 
of temperature of one degree between calorimeter and enclosure, 
calculated on the assumption that the cooling is proportional to 
the difference of temperature between the calorimeter and 
enclosure, an assumption which the result shews is only ap- 
proximately correct. 

If the mass of water in the calorimeter is M, then the heat 
lost will be Mh gramme degrees per minute, and if A is the 
total area of the calorimeter exposed, the heat lost per unit 
surface in one minute will be MhjA, or if k is the heat lost per 
sq. cm. of surface per second, per degree excess of temperature 

Calculate k from the mean value of h found, and enter your 
results as follows. For small differences of temperature k 
should only depend on the nature of the radiating surface. 

Inner radius of enclosure ... ... ... 6 cms. 

Weight of water in calorimeter ... ... 146 grms. 

Radius of calorimeter ... ... ... 3*2 cms. 

Height of „ ... ... ... 5"1 „ 

Area of curved surface of calorimeter . . . 102'6 sq. cms. 

„ top and bottom „ ... 64'4 „ 

Total area of radiating surface ... 167"0 „ 
Mean value of h found above = "0126. 

^ , 0126 X 146 .^^^i«^ 

^^^^^ ^= 60x167 =QQQ1^^' 

The inner radius of the enclosure is not required in the 
calculation, but should be measured and recorded, in order that 



XXIV LAWS OF COOLING 123 

the effect of the dimensions of the air space on the heat lost 
by conduction and convection might be traced by comparing 
the results obtained under different conditions. 

Newton's Law of Cooling states that the rate of loss of 
heat of a hot body is proportional to the difference in tempera- 
ture between the surface of the hot body and the surrounding 
space. The above experiments have shewn that the law is 
only approximately true, but it holds with sufficient accuracy 
in heat measurements where the differences in temperature do 
not exceed a few degrees. 

In order that the results obtained in the above experiments 
may be utilised in calorimetric measurements, a few further 
explanations are necessary. 

If h is the cooling in a given interval of time of a calorimeter 
containing a liquid, the specific heat of which does not vary 
with the temperature, when the difference in temperature 
between the calorimeter and the enclosure is one degree, 
Newton's law states the rate of cooling H when the tem- 
perature of the enclosure is T^ and that of the calorimeter 
r to be 

H^h{T-T,). 

If Tq is known, the value of h may be determined from a 
single experiment, as in Exercise II. above. 

It is often impossible to know the value of T^ accurately, as 
different parts of the enclosure may have different temperatures. 
In this case the above equation will still hold, but Tq will 
represent the mean temperature of the enclosure. To deter- 
mine h we then require two sets of observations with different 
values of T, but the same T^. Let H^ be the cooling of the 
calorimeter in a certain interval when the temperature is 2^, 
and H^ when the temperature is T^ ; then 

H,^h{T,-T,) and H^^h{T,--T,). 

As If,, ifg, Tx, T^ are known quantities, the values of h and 
T^ may be deduced and substituted in the general equation 

H^h{T"T,), 



124 HEAT XXIV 

It is found in this way that 

and ,,^^^-r)-^.(r.-r) 

which gives the fall of temperature in the interval, for any 
value of T. 

Sufficiently accurate values may be obtained by a graphical 
method. Let OAo (Fig. 47) represent the temperature To of the 
enclosure, and OAi the temperature Tj of the calorimeter for 
which the cooling has been determined. Let AiN^ represent 
the rate of cooling measured on some suitable scale. Join -^o-A^i- 
Then if OA be any temperature T, the corresponding rate of 
cooling will be measured by AN. For 

AN AAo _ jT — To _ Rate of cooling at temperature T 
AiNi ~ AjAq ~ T^ — Tq ~~ Rate of cooling at temperature T^ ' 

Hence, if a curve is drawn such that the ordinates measure 
rates of cooling, and the abscissae 
the corresponding temperature of 
the cooling body, the curve is a 
straight line within the limits of 
accuracy of Newton's Law of Cool- ^-"^^^o 

ing. Fig- ^7. 

If the temperature T^ is not known, but the rates of cooling 
H^ and H^ are found for the two temperatures T^ and Tg, we 
may deduce the rate of cooling for a temperature T as follows : 

Take OA^ and OA^ (Fig. 48) to represent T^ and T^, A^N^ 
and A^Nc^ the rates of cooling H^ and H^. Join N-^N^^. If OA 
be any temperature T, then AN will be the corresponding 
rate of cooling. 

It may happen that at the beginning of an experiment, the 
water in the calorimeter at a temperature T^ does not fall but 
rises in temperature. In that case the line AiN-^ must be 
measured downwards as in Fig. 49, i.e., it must be taken as 
negative cooling. 

In drawing the curves, attention should be given to the 




XXIV 



LAWS OF COOLING 



125 



scale. If the rates of cooling are measured to one per cent., 
for instance, which in all cases will be sufficient, the length of 





Fig. 49. 



the line A^N^ should be such that the hundredth part of it may 
be estimated, i.e., the unit in which it is measured should be 
chosen so that A^N^ is at least equal to ten times the length of 
a division on the squared paper used. 

The scale of temperatures may conveniently be such that 
the angle between A^A^ and N^N^ is between 30° and 45°. 

From the results obtained in Exercise II., making use of the 
observed rates of cooling in the two cases in which the differ- 
ences in temperature between the enclosure and calorimeter is 
least, determine, by the graphical method just described, the 
temperature of the enclosure, supposed not to be known. Thus, 
in the above example, one would take the abscissae proportional 
to 27*6 and 197 and the ordinates in some convenient scale 
proportional to 158 and 56. The line drawn through the two 
points so determined, intersects the axis of temperatures at the 
point representing the temperature of the enclosure. Carry 
out the construction for your own experiments, record the 
value of To thus found, and compare it with that obtained by 
direct observation. In order that jTo should be capable of being 
determined independently, it is clearly necessary that it should 
not vary during the course of the experiment. This is the 
reason why in Exercise 11. the temperature of the enclosure 
was adjusted so as to be the same in each case. 



SECTION XXV. 

COOLING CORRECTION IN CALORIMETRIC 
MEASUREMENTS. 

Apparatus required : Calorimeter with water jacket, ther- 
mometer graduated in tenths of degrees, india-rubber and heater. 

This exercise is an application of the principles explained 
in the exercise on the " Laws of Cooling," to the determination 
of the specific heat of india-rubber. India-rubber is a bad 
conductor of heat, and in consequence gives up its heat to the 
calorimeter at a slow rate; hence the correction for cooling is 
of special importance. 

The object of the exercise is not so much to obtain an 
accurate value of the specific heat as to shew the principle 
of reducing calorimetric observations. 

Place the india-rubber in a heater, and raise it to about 
100° C. This may be done with sufficient accuracy by placing 
it in boiling water for not less than a quarter of an hour. 

While the india-rubber takes up the proper temperature, 
weigh the calorimeter and fill it and the water jacket with 
water which has previously been standing in the room, so as to 
take up its temperature — tap water is generally much colder. 
Weigh the calorimeter again to determine the quantity of 
water in it. 

Place the calorimeter with the thermometer in it in its 
water jacket, and leave it for two or three minutes before 
taking any observations. In the meantime prepare your note- 
book by drawing vertical lines to allow for five vertical columns. 
The first column is reserved for the insertion of the time at 
which an observation is taken, which should be carefully put 
down; the second is for the temperature observed. 



XXV COOLING CORRECTION 127 

The experiment is divided into three stages, during which 
observations of temperature are taken every half minute. 

1. The initial period, before the introduction of the india- 

rubber into the calorimeter. This should last six 
minutes, and should only be begun when the change 
of temperature is small and steady. Observations are 
taken at half-minute intervals during two minutes, 
then stopped during two minutes and resumed again. 

2. The principal period, in which the india-rubber is in 

the calorimeter. The thermometer will be found at 
first to rise, then to remain nearly steady, and then to 
fall. When the fall has become uniform we arrive at : 

3. The final period, of six minutes, during which the fall 

is steady. Observations are taken as in the first 
period. 

About 20 seconds after the last reading of the first period, 
the india-rubber is removed from the boiling water, the adhering 
water shaken off, and at the half minute the rubber is lowered 
into the calorimeter. 

During the ten seconds that the india-rubber is exposed to 
the air the water on its surface evaporates, and is not carried 
into the calorimeter. 

In an accurate measurement of specific heat this would 
not be a permi.ssible process, as the parts of the india-rubber 
exposed to the air are cooled by evaporation of the water and 
by conduction to the air. The substance would have to be 
heated in an air chamber kept at 100° C. by being surrounded 
by steam, and be suddenly dropped into the calorimeter. 

When the india-rubber has been placed in the calorimeter, 
the thermometer is observed at the next half minute and each 
succeeding half minute. The first few temperature observations 
will only be approximate, as the temperature will be rising 
quickly. 

Observation of temperature should be made till it has ceased 
to rise and has fallen at a constant rate for six minutes. 

In order to reduce the observations and calculate the specific 
heat, deduce from the observations taken during the first and 
third periods the rates of cooling or heating at the mean 



128 



HEAT 



XXV 



emperatures of the two periods, and determine graphically, as 
explained in the previous section, the rate of cooling corre- 
sponding to any temperature of the calorimeter. 

Enter^ in the third column of the note-book the mean 
temperature of the calorimeter during each interval of time 
throughout the experiment. Determine from the construction 
and put down in the fourth column, the diminution of tempe- 
rature owing to the loss of heat during each interval. In the 
fifth column tabulate the total loss in temperature up to each 
time given in the first column. An inspection of the table 
below will shew how this column is obtained. In the example 
given the loss was "005 to the end of the first interval, and 
during the second interval the loss was "009 ; the total loss up 
to the end of the second interval was therefore "014, and if to 
this is added the loss during the third interval we obtain '025 
and so on. 

The last column, which gives the temperature which the 
calorimeter would have shewn if there had been no loss of heat 
due to radiation and conduction to the air, is obtained by adding 
the numbers in the fifth column to those in the second. 

It will be seen that the temperatures given in this column 
rise at first quickly, then slowly, and during the last period 
remain sensibly constant. The mean corrected temperature 
during that period would therefore have been 22*54'', if there 
had been no loss of heat. 



First Period. 

13 March, 1899. 

Temperature slowly falling. 



Time 


Temp. 


Time 


Temp. 


Cooling 


ll'^OO-O'" 


19°-69 


111^4-0™ 


19°-67 


•02 


0-5 


■69 


4-5 


•67 


•02 


10 


•69 


5^0 


-m 


1 -03 


1-5 


•68 


5-5 


•66 


1 ^02 


2 


•67 


6-0 


•66 


•01 


Means 


19*'-684 




19°-664 


•020 



XXV 



SPECIFIC HEAT OF INDIA-RUBBER 



129 



The mean of 19*684 and 19664 gives the 

Temperature at ll*' 3°» = 19°-674 
Cooling in 4 mins. = '020 
i .. ='0025 
3i „ = 0°018 

Temperature at ll'^e^'S) _ jqo.^-p 
when rubber introduced] ~ 

Second Period.* 





Temp. 


Average temp 
during 
Interval 


Cooling 

in 
Interval 


Total loss 


Corrected 
temp. 


11 h. 07-Om. 


20-85° C. 


20-26 


-005 


-005 


20^86 


07-5 


21-30 


21-08 


•009 


-014 


2131 


08-0 


21-60 


21-45 


•Oil 


•025 


21 ^62 


08-5 


21-77 


21-68 


•012 


•037 


21-81 


09-0 


21-90 


21-84 


•013 


•050 


21-95 


09-5 


22-01 


21-96 


•014 


•064 


22-07 


10-0 


22-08 


22-04 


-014 


•078 


22-16 


10-5 


22-17 


2212 


-014 


•092 


22-26 


11-0 


22-20 


22-18 


-015 


-107 


22-31 


11-5 


22-23 


22-22 


•015 


•122 


22-35 


12-0 


22-26 


22-24 


-015 


•137 


22-40 


12-5 


22-28 


22-27 


-015 


•152 


22-43 


13-0 


22-30 


22-29 


-015 


•167 


22-47 


13-5 


22-31 


22-30 


•015 


•182 


22-49 


14-0 


22-32 


22-32 


•016 


•198 


22-52 


14-5 


22-32 


22-32 


•016 


•214 


22-53 


150 


22-31 


22-32 


•016 


•230 


22-54 






Third Period.* 








Temperature falling steadily. 








Temp. 


Average 
temp. 


Cooling in 
Interval 


Total loss 


Corrected 
temp. 


11 h. 15-5 m. 


22-30° C. 


22-31 


-015 


•245 


2254 


16-0 


22-28 


22-29 


•015 


•260 


22-54 


16-5 


22-27 


22-28 


•015 


•275 


22-54 


170 


22-26 


22-26 


•015 


•290 


22-55 


17-5 


22-24 


22-25 


-015 


•305 


2254 


180 


22-22 


22-23 


-015 


•320 


22-54 


18-5 


22-20 


22-21 


•015 


•335 


22-54 


190 


22-20 


22-20 


-015 


•350 


22-55 


19-5 


22-19 


22-20 


•015 


•365 


22-56 


20-0 


22-17 


22-18 


•015 


•380 


22-54 


20-5 


2215 


22-16 


-015 


•395 


22-55 


210 


2213 


22-14 


•014 


•409 


22-54 


21-5 


2211 


22-12 


•014 


•423 


2264 








Mean during last i^eriod 
Initial temperature ... 


22-544 
19-656 



Riae of temperature 
* 8m Note at the end of the Section. 



2-888 



8. P. 



130 HEAT XXV 

Calculat{o7i of Rate of Cooling during third period. 





22-30 - 


■ 22-19 


= 


11 








22-28 - 


- 22-17 


= 


11 








22-27 - 


2215 


= 


12 








22-26 - 


- 22-13 


= 


13 








22-24 - 


- 22-11 
Mean 





13 

120 






Hence cooling 


in 4 minutes 


= 


0° 


-120 




)j 


» 2 5> 




= 


0° 


-015 



This cooling corresponds to a mean temperature of 22°*2 
which is obtained with sufficient accuracy from the temperature 
observed at the middle of the third period. 

In an accurate determination of specific heats we are able 
to measure a rise in temperature of a few degrees to about one 
part in a thousand, although the observations have only been 
taken to the hundredth part of a degree. The increased ac- 
curacy is obtained by making use in the initial and final stage 
of a number of observations and taking the mean. In the 
present instance, the initial temperature of the india-rubber, 
owing to its treatment, must be doubtful to one degree, and 
the l£ist figure need not be taken into account in the final 
calculations, although it should be worked out as in the 
example given for the sake of practice. 

Knowing the masses of india-rubber and water, and the 
water equivalents of the thermometer and calorimeter, the 
specific heat of india-rubber can now be calculated. 

The final results are entered as follows : 



Weight of india-rubber 


= 1264 grams 


calorimeter 


= 28-14 „ 


„ „ and water 


= 155-9 


Mass of water 


= 127-8 


Calorimeter and thermometer equivalents 


= 3-2 


Total water equivalent 


= 1310 


Initial temperature of india-rubber 


= 100° C. 


Final 


= 22°-5 


Fall in temperature „ „ 


= 77°-5 


Rise of temperature of water 


= 2°-89 


Hence specific heat of india-rubber 


= 0-387 



XXV 



SPECIFIC HEAT OF INDIA-RUBBER 



131 



Draw curves (Fig. 50) shewing the observed changes of tem- 
perature and the temperature curve after correcting for cooling. 



jal 



-^^ 
^ 



T^ 



:& 



3E 



as 



ie 



i£ 



Tlm« 

-XL 



Fig. 60. 

The circumstances under which the loss of heat has been 
determined during the first and third periods are not strictly 
the same. At first the india-rubber was not immersed, and 
therefore the water equivalent was different, and also the level 
of the water in the calorimeter. This intnxiuces an error in 
the cooling correction which is very small when the change of 
temperature during the first period is small. 

Note. When unnecessary decimals are discarded, it is an elementary 
rule to increase the last remaining figure by one, when the first discarded 
figure is higher than five or five followed by other figures. But when only 
one figure is rejected, that figure being five, some doubt may arise, as to 
whether to leave the last figure as it stands or to increase it by one. We 
should commit a systematic error, if in the last column of the table on 
page 129, we were uniformly to take either the higher or the lower estimate. 
An excellent rule, adopted in the United States and deserving to come into 
general use, is always to leave the last figure an tvtn number. Thus 2535 
and 30*65 should be shortened into 25*4 and 30*6. 

9—2 



SECTION XXVI. 



SPECIFIC HEAT OF QUARTZ. 



Apparatus required: Calorimeter with water jacket, 
thermometer graduated in tenths of degrees, piece of quartz, 
steam-jacket heater. 

If the specific heat of a substance is to be determined 
accurately, the method of heating adopted in the previous 
exercise should not be used, since it results in an uncertain 
amount of hot water being carried over with the heated body 
into the calorimeter. The body to be heated is suspended by 
a thread in an air chamber warmed by steam or hot water, and 
allowed to remain till the reading of a thermometer placed in 



I 




Fig. 51. 



XXVI SPECIFIC HEAT OF QUARTZ 133 

the chamber has been constant sufficiently long to secure 
uniformity of temperature throughout the heated body. 

Fasten about 50 cms. of thread to the piece of quartz 
provided, and suspend the quartz in the centre of one of the 
steam-jacket heatere (Fig. 51). 

Measure the diameter of the thermometer, the length of its 
bulb and of that part of its stem which is immersed in the 
water of the calorimeter ; hence calculate the volume to about 
'5 cc. Assuming it to be all mercury, calculate the water 
equivalent. As the heat capacities of equal volumes of glass 
and mercury are not greatly dififerent, the equivalent of a 
thermometer may often be calculated in this way with sufficient 
accuracy. 

While the substance is heating fill the calorimeter about 
two-thirds full of water at the temperature of the room, the 
quantity of water used being found by weighing the calorimeter 
before and after filling. When the thermometer in the heater 
has been steady for 15 minutes, commence to take the observa- 
tions of temperature of the calorimeter for the " First period," 
as described in the previous section. 

At the end of this period, place the calorimeter and jacket 
under the heater and let down the quartz into the calorimeter 
as quickly as possible. 

Replace the calorimeter in its original position, take obser- 
vations during the " Second " and " Third " periods, and thence 
calculate the specific heat of quartz as in the previous section. 

Calculate the specific heat also by the following approximate 
method: determine the rates of change of temperature per 
interval of time during the first and third periods, and thence 
deduce the rate of loss at the temperature, which is the mean 
of the temperatures at the commencement of the second period, 
and of the maximum temperature attained. 

Multiply this rate of loss by the number of intervals between 
the commencement of the second period and the time at which 
the maximum temperature is attained, and add the product 
to the maximum temperature. This will, if the time is short, 
i.e. if the substance gives up its heat rapidly to the water of 
the calorimeter, be approximately the temperature which the 



134 HEAT XXVI 

calorimeter would have attained if there had been no loss of 
heat. 

Record the calculation as follows : 

10 Nov. 1898. 

Rate of cooling during first period = 0°*01 per half minute. 
Temp, on dropping quartz in ... =19°*72C. 
Maximum temp, attained ... = 22°"99 

Interval between dropping quartz] _ « i ir • . 
and max. temp. ... ...J 

Rate of cooling during third period = 0°-025 per half min. 

Mean rate of cooling = 0°-l 7 5 per half min. 

.-. Loss of temp. ... = '0175 x 3 = 0°-052 

Corrected maximum temp. ... = 23°04 

Rise of temp. ... ... ... = 3°*32 

Water heated + water equivalent =181 grams. 

.'. Heat absorbed by water ... =601 gram-degrees. 

Initial temp, of quartz ... ... = 99°'8 

Final „ „ =23°-0 

Decrease of temp. ... ... = 76°"8 

Weight of quartz = 41*2 grams. 

. • . Specific heat of quartz = —^ — =-^ = 191. 

It is instructive to compare the results obtained by this 
approximate method of treating the cooling with the more 
accurate one given in the last section. It will be found that 
even in the case of a substance taking up the temperature as 
quickly as quartz the difference is appreciable. 



SECTION XXVII. 

LATENT HEAT OF WATER. 

Apparatus required : CaloHmeter, thermometer graduated 
in tent f IS of degrees, piece of ice. 

The latent heat of water, i.e. the heat absorbed when 1 gram 
of ice melts, may be determined by adding ice at 0°C. to 
sufficient water to melt it completely, and determining the 
decrease of temperature of the water. 

Weigh the calorimeter provided, place in it about 170 grams 
of water at about 18° C, and weigh again. 

Select a piece of ice weighing about 10 grams. 

Place in the water a thermometer gi-aduated to tenths of 
degrees, and observe the temperature of the water every half 
minute for 6 minutes (" First period ") as explained in Section 
XXV. 

Dry the surface of the ice thoroughly with blotting paper, 
and at the end of the next half minute drop it into the water 
of the calorimeter. By means of the stirrer keep the ice under 
water and the water stin*ed. Take half-minute observations of 
temperature till it reaches a minimum and then commences to 
rise — " Second period." Continue observations of temperature 
for 6 minutes after the rise has become uniform, these obser- 
vations constituting the ** Third period." 

From the observations taken in the first and third periods 
plot the curve from which the cooling at any temperature can 
be found, and apply the correction for cooling (or in this case 
heating) as described in Section XXV. 

Weigh the calorimeter to determine the amount of ice 



136 



HEAT 



XXVII 



which has been added, and from your observations determine 
the latent heat of water. 

Work out from elementary principles the formula applicable 
in this case and record as follows : 



10 January, 1899. 

Initial temperature of calorimeter . , 

Final corrected „ „ 

Fall of temperature ... 

Weight of calorimeter and stirrer . . 

„ cal. + water 

„ water 

„ cal. + water + ice 

„ ice added 
Water equivalent of calorimeter 

„ „ thermometer .. 
Total,, „ 



= 16°-48C. 

= 12°-24 

= 4°-24 

= 59 '1 grams. 

= 2300 

= 170-9 

= 238 2 

= 8-2 

= 5-3 

= '5 

= 176-7 



T ^ ^i. X 176-7 X 4-24 ,_^. ^„ , 
.-. Latent heat = ^^ 12*24 = 79-1. 



SECTION XXVIII. 



LATENT HEAT OF STEAM. 



Apparatus required ; Calorimeter, condenser, thermometer ^ 
flash, stand and burner. 

The latent heat of steam, i.e. the heat necessary to convert 
water at 100'^ C. into steam at the same temperature, may be 
determined by condensing a known weight of steam and 
observing the heat given up during the process to the liquid 
of the calorimeter, generally water, kept always below 100" C. 
The condensation takes place more regularly if the steam is 
allowed to condense in a separate vessel and not in the water of 
the calorimeter itself. 



f^ 



A-r-^ 




Fig. 52. 



Fig. 68. 



Weigh the calorimeter stirrer and condenser (Fig. 52) 
provided, first empty, then remove the condenser, weigh it to 



138 HEAT XXVIII 

•01 gram, replace, fill the calorimeter to within about 2 cms. of 
the top with water at the temperature of the room, and 
again weigh. 

Arrange the vessel in which the steam is to be generated, 
the delivery tube and the calorimeter as shewn in Fig. 53. 

When this has been done, disconnect the tube from the 
condenser and place a burner under the boiler, regulating the 
height of the flame so that the water boils gently, the steam 
being allowed to escape from the end of the tube. 

Observe the temperature of the water in the calorimeter for 
6 minutes — the " First period " of previous sections. Then 
replace the calorimeter, and insert the delivery tube into the 
head of the condenser tube so that the steam passes into the 
condenser and is condensed there. Keep the water well stirred, 
and observe the temperature every half minute, "Second period," 
till it has been raised about 10° C, then remove the delivery 
tube from the condenser. 

Continue observations of temperature till the change of 
temperature has been uniform for 6 minutes, " Third period." 

Weigh the calorimeter and contents. 

Determine the cooling during the first and third periods 
and by means of the values found correct the observed 
temperature throughout the experiment for cooling, and find 
the latent heat of steam at the boiling point corresponding 
to the atmospheric pressure at the time. 

Work out the necessary equation and record as follows : 



7 December, 1899. 




Barometer 7 5 '3 cms. Boiling point of 




water 


= 100° - -37 X 7 




99°-74 


Weight of calorimeter, stirrer and condenser 


91-82 grms. 


„ „ „ „ condenser and 




water ... 


223-24 „ 


.-. weight of water 


131-42 „ 


Water equivalent of calorimeter, stirrer, 




condenser and thermometer 


8-8 „ 


. • . total water equivalent 


140-2 „ 



XXVIII LATENT HEAT OF STEAM 139 

Initial temperature ... ... ... 14°*76 

Final temperature corrected according to 

method of Section XXIV 32°-12 

Rise in temperature ... ... ... 17°*36 

Final weight of water, calorimeter, etc. ... 22714 grms. 

Weight of steam condensed 3*90 grms. 

Latent heat of steam = — — ^^- 82*2 = 542 

Experiments on latent heat of vaporisation are liable to a 
number of errors owing to the difficulty of taking account of 
the gain and loss of heat at the point where the steam is led 
into the apparatus. Hence the results obtained, when the 
experiments are made on a small scale are very uncertain. 
The above represents an average determination with the 
apparatus used. 



SECTION XXIX. 

HEAT OF SOLUTION OF A SALT. 

Apparatus required : Small calorimeter with suspending 
hook, larger calorimeters, thermometers, and salts. 

If p grams of a salt, the molecular weight of which is m, be 
dissolved in P grams of a solvent of molecular weight M, the 
solution formed has pjm gram molecules of the salt to PjM 
gram molecules of the solvent, or 1 gram molecule of the salt to 
every n = PmjpM gram molecule of the solvent. 

If the specific heat of the solution formed be c, and if 
during the process the temperature of the solution decreases 
from to to t, the quantity of heat absorbed by the solution of 
the salt is 

[{P^p)c-^w][t,-t], 

where w is the water equivalent of the calorimeter and thermo- 
meter. The quantity 

{ {P+p)c + w]{t,-t] 
^ P 

is the heat of solution of 1 gram of the salt, and the quantity 

is the heat of solution of 1 gram molecule of the salt, and is 
called the " molecular heat of solution." 

The molecular heat of solution of a salt is nearly constant 
for weak solutions, but diminishes as a rule as the strength of 
the solution increases. 



XXIX 



HEAT OF SOLUTION OF A SALT 



141 



Determine the molecular heats of solution of Sodium 
Chloride and of Ammonium Chloride in water by mixing 

20 grams NaCl (m = 58'5) in 98'5 grams water (M— 18), n — 16 
20 „ „ „ „ 147-7 „ „ „ 71 = 24 

15 „ NH4Cl(m = 53-5)inl01 „ „ „ n = 20 



li 



151 



71 = 30 



The specific heats of the solutions may be taken as 84, '87, '87 
and 89* respectively. 
Proceed as follows : 

Place the requisite quantity of water at about 18° '5 C. in a 
calorimeter surrounded by an air space and water jacket at the 
temperature of the room. 

Weigh the salt, put it into one of the small calorimeters, 
and suspend it by means of its hook in the water of the large 
calorimeter. Place a thermometer graduated to tenths of a 
degree in the water. After about 10 minutes take observa- 
tions of temperature for 6 minutes. If the change of tempera- 
ture is regular, unhook the small calorimeter and upset it in 
the water so that the salt and water come into contact with 
each other. Stir the mixture well and observe the temperature 
every half minute till the change has been regular for at least 
6 minutes. 

From your observations determine the molecular heat of 
solution in each case, recording in the usual way and making 
the proper corrections for cooling. 

Tabulate your results as follows : 









Molecules H,0 




( 


t 






8«lt 


p 


P 


to lott^ 

••It 


to 


observed 


oorrected 


Q 


Q 


NaCl 


20 


98-5 


16 


18'-12 


16*14 


1405 


11-4 


660t 


)f 


20 


147-7 


24 


18-05 


16°'48 


16-38 


13-4 


770 


NHCl 


15 


101 


20 


2r-32 


13°-21 


1301 


60-5 


3200} 


»> 


15 


151 


30 


2rio 


14°-43 


14-27 


72-6 


3900 


« 


Von Buchka, TabelUn, pp. 275, 276. 


1 


H Berthelot, Therviochitnif, ii. p. 202. 






1 


^ Ibid. p. 222. 













SECTION XXX. 

THE MECHANICAL EQUIVALENT OF HEAT OR 
SPECIFIC HEAT OF WATER IN WORK UNITS. 



Apparatus required: Pulujs friction cones with rotating 
pulley, jar of water, float, thermometer. 

When a gram degree of heat, i.e. the heat necessary to raise 
1 gram of water at 15° C. to 16° C, is generated by the per- 
formance of mechanical work, the work done is called the 
mechanical equivalent of heat or the specific heat of water in 
work units (ergs per degree). 

To determine this quantity, the work may be done in a 
variety of ways; the one adopted in what follows depends on 




Fig. 54. 



one solid being made to slide along another against friction. 
In order that the sliding motion may be continuous the solids 



XXX MECHANICAL EQUIVALENT OF HEAT 143 

are circular, one, a small hollow cone of steel, fits into another 
similar cone slightly larger. The lower outer cone is held in a 
frame which can be set in rapid rotation about a vertical axis 
coincident with that of the cone. The smaller cone is filled 
with mercury and is placed in the rotating cone but is pre- 
vented from rotating by a light wooden arm. To one end of 
this arm a thread is attached which passes over a pulley and 
carries a float placed in a jar of water. 

The moment of the couple which the tension in the thread 
exerts on the inner cone is equal and opposite to that which 
the rotating outer cone exerts on the inner cone. The work 
done by the frictional couple in any interval of time is equal to 
this moment multiplied by the angle through which the outer 
cone has in the interval been rotated with respect to the inner 
cone. To determine this angle the apparatus is provided 
with two dial wheels which register the number of revolutions 
of the outer cone. The angle of rotation is 27r times the 
number of revolutions. 

The tension in the thread is equal to the effective w^eight 
of the float which is numerically equal to ^r x volume of float 
pulled out of the water. 

Take the two cones out of the supporting frame, see that 
their surfaces are clean and weigh them together. Weigh the 
screws by which the wooden pointer is attached to the inner 
cone. Fill the inner cone to within 3 mms. of the top with 
clean mercury and weigh again to get the weight of mercury. 
Taking the specific heat of the steel of the cones to be 'llO, 
that of mercury to be '033 and that of brass '09, calculate the 
water equivalent of the cones and contents. 

Replace the outer cone in the supporting frame, taking care 
that it does not touch any of the metal of the frame. By 
means of the adjusting screws at the sides of the frame, centre 
the cone accurately so that it revolves about its own axis. 
Attach the wooden pointer to the inner cone and place the 
cone in the outer as in the figure. Adjust the position of the 
float cylinder and the length of the thread so that when the 
cone spindle is rotated at a convenient speed the thread and 
the wooden rod are perpendicular to each other. See that 



144 HEAT XXX 

readings of the surface of the water can be taken on the 
graduated scale of the float, both when it is raised by the 
tension of the thread, and when the thread is quite slack. 
The difference between the two readings should be 5 to 10 cms. 
This can be secured by using a little petroleum as lubricant 
between the cones if the difference is too gi-eat, and a little 
vaseline if it is too small. Take the reading of the float when 
the tension in the thread is zero. Read the two dials attached 
to the rotating apparatus, and measure the length of the 
wooden rod from the centre of the cones to the point of attach- 
ment of the thread. 

Measure the diameter and length of the bulb of the thermo- 
meter and calculate its water equivalent. Hang the thermometer 
on the movable arm attached to the stand and lower the bulb 
into the inner cone till it is below the level of the mercury. 

Take observations of temperature every half minute for 
3 minutes, then wait 3 minutes and take observations from 
3 more minutes. At the end of this interval commence 
to rotate the hand wheel steadily, continuing to observe the 
temperature every half minute. At the middle of each half 
minute take a reading of the surface of the water on the float. 
Continue the rotation till the temperature has risen about 5°C. 
Then stop the rotation, read the temperature till the change 
has been regular for 3 minutes, wait 3 minutes, and read again 
for 3 minutes. Read the dials and the float. 

Remove the float from the water and measure its cross- 
section at 3 or 4 points, where the readings have been taken. 
Take the mean of these measurements. Take also the mean of 
the readings of the float in the raised position, and subtract 
from it the mean reading for no tension. The product of this 
difference into the mean cross-section of the float is the mean 
tension in gravitation measure. Multiply this by Stt times the 
number of revolutions and by the length of the wooden arm and 
by g, and the product is the work done during the rotation in ergs. 

Apply the correction for cooling to the temperature readings 
as in Section XXV. and determine the corrected rise of tem- 
perature. The product of this by the water equivalent of cones 
and contents is the heat generated in gram-degrees, i.e. in terms 



XXX 



MECHANICAL EQUIVALENT OF HEAT 



145 



of the unit of heat to which the specific heats used have been 
referred. 

The quotient of the number of ergs of work done by the 
number of gram-degrees of heat generated is the number of 
ergs work required to generate heat sufficient to raise 1 gram 
of water 1 ° C. 

Arrange your observations and results as follows : — 

10 June, 1899. 

Weight of steel cones 

., „ and mercury 

„ mercury 

Water equivalent of cones . . . 84'4 x 

„ „ mercury ... 149-2 x 

thermometer ... 

Total water equivalent of cones and contents 

Reading on stem of float without tension = 

Readings during rotation : — 
19-70, 19-60, 19-55... &c., mean 
Mean rise of float ... 
Initial reading of counter ... 
Final „ ., 

No. of revolutions ... 
Angle turned through 27r x 1765 
Length of arm of lever 
Cross-section of float stem 

= 102, 1-01, 1-00, -98, -99, mean 
.-. Work done = 257 x 13 x 1 x 11090 x 981 = 363 x 10' ergs. 

Temperature at end of first period 15'' 00 C. 

Mean temperature at end of third period) on^.?' 

after coiTecting for cooling J 

Rise of temperature ... ... ... ... 5°'75 

Heat generated = 1493 x 5 75 = 858 gram degrees. 

363 X 10' 
858 ' 



= 84*4 grams. 

= 233-6 „ 

= 149-2 „ 

119 = 9-76 
033 = 4-97 
... = 10 

... = 14-93 
= 65 2 cms. 

= 19-52 „ 
= 130 „ 
= 105 
= 1870 
= 1765 

=11090 radian.^. 
= 257 cms. 

= 100 sq. cm. 



Mechanical equivalent = 



= 42 '3 X 10* ergs per degree. 



8. p. 



10 



BOOK IV, 

SOUND. 



SECTION XXXI. 

FREQUENCY OF A TUNING FORK BY THE SYREN. 

Apparatus required : Tuning fork, singing flame, syren 
and blowing apparatus. 

To enable the comparison of a fork and a syren to be made 
more conveniently than it can be done directly, it is usual to 
tune a singing flame to the fork by adjusting the length and 
position of the resonating tube over the flame, and then to tune 
the syren to the flame. 

Calculate the length of an open pipe which will act as a 
resonator to the fork, the vibration frequency of which is 
supposed to be known roughly. Take a glass tube of rather 
less length, and roll a piece of paper round one end so that the 
effective length of the tube may be altered by sliding the paper 
tube along. Place the tube above a small gas flame produced 
by gas issuing from a minute hole at the end of a conical glass 
tube. 

Bring down the pipe on to the flame, and adjust till the 
flame " sings " ; then vary the position of the paper tube till 
the note emitted by the pipe produces no beats with the note 
of the fork (Fig. 55). 



XXXI FREQUENCY OF A TUNING FORK BY THE SYREN 147 

Place the syren on the blowing apparatus, start the appa- 
ratus and increase the rate of blowing till the note emitted by 
the syren produces no beats with the pipe. 




Fig. 65. 

Maintain the rate of blowing, and at a given instant take a 
reading of the positions of the fingers on the dials indicating 
revolutions of the spindle of the syren. 

At the end of a minute again take readings. Subtract the 
readings to get the number of revolutions in the interval, count 
the number of holes in the disc of the syren, find the product 
and divide by 60, the result is the frequency of the note of the 
syren, and hence of the pipe, and fork. 

Tabulate your results as in previous exercises. 



10—2 



SECTION XXXIL 



VELOCITY OF SOUND IN AIR AND OTHER BODIES 
BY KUNDT'S METHOD. 



Apparatus required : Kundt's apparatus, rods and rubber. 

Kundt's apparatus consists of a glass tube of about 200 cms. 
length and 5 cms. diameter into one end of which a rod of wood, 
metal or glass projects (Fig. 56). 



^r- 




Fig. 56. 

The rod is securely clamped to the table at its middle point, 
and can be set into vibration parallel to its length by stroking 
it, if wood with a piece of cloth on to which a little resin has 
been rubbed, if metal with a leather rubber similarly treated, 
or if glass with a damp cloth. The end which projects into the 
tube is provided with a cardboard disc which has a diameter a 
little less than that of the inside of the tube. 

The tube contains a little lycopodium powder or finely 
divided silica and the further end is stopped by a movable 
plug. It is important that both tube and powder should be 
quite dry. 

When the rod is set into logitudinal vibrations, the disc on 
the end in the tube sets the air in the tube in vibration and 
the powder is carried along with the air. If the tube is long 
enough, there are certain parts of it where the air is not in 



XXXII VELOCITY OF SOUND IN AIR AND OTHER BODIES 149 

motion along the tube, and at these parts the lycopodium or 
silica remains in heaps undisturbed. 

These heaps therefore indicate the positions of the nodes of 
the vibrating column of air, and twice the distance between 
consecutive heaps is the wave length of the vibration in the 
gas witlj which the tube is filled — in this instance air. 

Since the rod is clamped at its centre, this point will be 
a node in the rod, and the two ends will be the centres of 
vibrating segments, so that the wave length of the vibration in 
the rod is twice the length of the rod. 

Since the frequencies of the rod and of the gas in the tube 
are identical, the ratio of the distance between the two heaps of 
lycopodium or silica to the length of the rod, is the ratio of the 
velocities of sound in the gas and in the material of the rod. 

Determine by this method the velocity of sound in brass 
and in glass, given the velocity of sound in air, and compare 
the velocities of sound in air and in coal gas. 

Clean and dry the tube provided, and scatter in it a small 
quantity of finely divided silica. Clamp the brass tube at the 
middle point to the support, and clamp the support to the 
table. Slide the tube over the rod and support it on blocks so 
that the cardboard disc at the end of the rod can move in the 
tube freely. Tap the tube sharply so that the silica collects in 
a line along the tube. Rotate the tube about its axis through 
a few degrees, so that the line of silica is not at the lowest 
point of the tube. Rub the rod lengthwise with a piece of 
leather on which resin has been rubbed, watching the line of 
silica during the process. If no motion is perceived, move the 
plug at the further end of the tube a centimetre and repeat. 
Continue till a position of the plug is found for which, when the 
the rod is rubbed, some of the silica is blown along the tube 
and falls to the lowest point in a number of heaps. Move the 
plug by millimetres at a time till this action appears most 
energetic, and when the heaps are distinct and nearly touch 
each other, count their number and observe on the scale under 
the tube, the p<:)sition8 of three consecutive spaces between the 
heaps at each end of the tube. Take the mean of the first 
three and the mean of the last three, subtract and divide by 



150 SOUND XXXII 

the number of heaps between the mean readings, i.e. find the 
distance between consecutive heaps. Observe the temperature 
of the air in the neighbourhood of the tube, and measure the 
length of the brass rod. Substitute for the brass rod one of 
glass, and repeat the experiment, then one of wood and again 
repeat. 

Record as follows : 

21 January, 1897. 

Brass tube 129 "5 cms. long. 

Readings of centres of intervals between heaps of silica : 

3000 cms. 91-50 cms. 

42-30 103-80 

5470 116-20 



Means 42-33 10383 

Difference = 61*50 cms. for 5 intervals. 
Distance of heaps apart = 12-30 cms. 
Temp, of air... ... =18"C. 

Velocity of sound in brass / velocity of sound in air 

= 129-5/12-3 = 10-53. 
Velocity of sound in air at 18° C. = 341 metres per second. 
.-. velocity of sound in brass = 3590 metres per second. 

Similarly for the other rods. 

By fitting corks to the end of the tube and filling it with 
some other gas, the velocity of sound in that gas may be 
compared with the velocity in air. 



SECTION XXXIII. 

STUDY OF VIBRATIONS OF TUNING FORKS BY 
MEANS OF LISSAJOUS' FIGURES. 

Apparatus required : Two large tuning forks, one the 
octave of the other, lens and drop of mercury. 

When a tuning fork is set into vibration by one of the 
prongs being displaced from its normal position with respect to 
the other, both prongs are set into oscillation by the elastic 
forces which resist deformation of the fork. These forces depend 
mainly on the cross section of the prongs near their roots, while 
the masses which most influence the movement are those near 
the ends of the prongs. The influence on the frequency of a 
fork of a small mass added to a prong near one end, may be 
readily studied by the help of '' Lissajous' figures." 

Arrange the two large tuning forks provided, so that the 
directions of vibration of the prongs are horizontal and vertical 
respectively. To the end of the upper prong of the fork 
vibrating vertically, attach a small lens of about 7 cms. focal 
length, and to the lower prong a small weight to counter- 
balance the lens. Place the second fork in such a position 
that the outer end of one prong is seen through the lens 
distinctly (Fig. 57). Rub a little grease on the end surface 
of the prong, and rub a drop of mercury into the grease 
with the finger. The drop will break up into a number of 
smaller drops which will adhere to the surface. Move the fork 
till one of the drops is seen distinctly through the lens. Strike 



152 



SOUND 



XXXIII 



the prongs of the two forks with a rubber stopper in the hole of 
which a short wooden rod has been inserted to serve as a handle. 




Fig. 57. 



On looking through the lens at the bead of mercury, a bright 
looped line is seen which changes its shape more or less quickly 
according to the relative frequencies of the two forks. The 
time which the curve takes tu go through a cycle of changes 
should be approximately noted. The adjustable weights on the 
prongs of one of the forks should then each be moved slightly, 
say towards the ends of the prongs, and the observation of the 
time of a cycle repeated. If it is greater than the previous 
time, the sliding weights should again be moved in the same 
direction, if less, in the opposite, till the time the curve takes 
to go through its changes of form is too long for the vibrations 
to be maintained throughout it. The two forks may then be 
considered in unison. Measure the distances of the weights 
from the ends of the prongs, move them both 1 mm. and 
observe the effect on the time the curve takes to go through its 
changes of form. Repeat the displacement of the weights till 
as many observations as possible have been taken on each 
side of the position for unison. 



XXXIII STUDY OF VIBRATIONS OF TUNING FORKS 

Tabulate the observations as follows : 



153 



Distance of weights from 
ends of prongs 


Displacement 


Time for gain 

of 1 vibration, 

seconds 


Vibrations 

gained per 

second 


Upper 


Lower 


67 cms. 


6-8 cms. 


cm. 


Unison 





6-6 


6-7 


•1 


10-4 


•096 


6-5 


6-6 


•2 


8-4 


•120 


6-4 


6-5 


•3 


6-8 


•147 


6-3 


6-4 


•4 


50 


•200 


6-2 


6-3 


•5 


4-0 


•25 


61 


6-2 


•6 


3-48 


•29 


60 


61 


•7 


2-92 


•34 


6-7 


6-8 





Unison 





6-8 


6-9 


•1 


100 


•10 


6-9' 


70 


•2 


8-6 




7-0 


7-1 


•3 


7-2 




71 


(fee. 








7-2 










7-3 










&c 











Draw a curve taking distance of the weights from the 
position of unison as abscissae and the vibrations gained per 
second as ordinates (Fig. 58). 




Fig. 68. 



BOOK V. 

LIGHT. 



SECTION XXXIV. 

MEASUREMENT OF ANGLES BY THE OPTICAL 
METHOD. 

If a body undergoes an angular displacement we may 
measure the displacement by reflecting a ray of light from a 
mirror attached to the body. If the plane of the mirror is 
parallel to the axis of rotation of the body, the reflected ray 
will turn through an angle which is double the angular dis- 
placement of the body. It is often found more convenient to 
measure the displacement of a ray of light 
reflected from such a mirror attached to 
the body than to measure directly the 
angle through which the mirror turns ; al- 
though theoretically as great an accuracy 
may be obtained by either method. 

In (Fig. 59) let XO be a ray of light 
incident on the mirror LM \ let ON be the 
normal and OK the reflected ray, K being 
the point at which the reflected ray cuts a 
line drawn through X at right angles to 
the incident ray. If a is the angle between 
the incident ray and the normal, the angle XOK = 2a and 

hence 

irZ=0Xtan2a (1). 




Fig. 59. 



XXXIV MEASUREMENT OF ANGLES BY THE OPTICAL METHOD 155 



If now the mirror turns through an angle so that a + ^ 
becomes the angle between XO and the normal, we shall have 

A''Z = 0Xtan2(a + ^) (2), 

where K' is the point at which the reflected ray cuts ^^ in 
the new position of the mirror. From the two equations 
(2) and (1) we can determine 6 

20 — arctan -^^y "" arctan j-y (3). 

If the line AB is divided into equal divisions K'X and KX 
can be read off directly and hence 6 calculated from (3) after 
OX has been measured. 

In order to make this method of measuring angles a practical 
one, we must of course deal with a beam of light and not with 
a single ray, and in that case the position of K will be well 
defined on the scale only when the beam comes to a focus at 
that point. In order to have the best definition the whole 
mirror should be filled with light, and hence the width of the 
beam at X must be twice the diameter of the mirror. 

Let in (Fig. 60) AB be the divided scale, L a lens with 
CTosH wires at F which can be 
illuminated from behind, then 
if the distance of the cross 
wires from the lens is properly 
adjusted, a real image will be 
formed at a point K of the 
scala On the other hand if 
K is a division of the scale 
suflficiently well illuminated, 
an image of A*" will appear at 
the point F. This image may 
be magnified by an eyepiece, 
and the observer looking 
through what is now a tele- 
scope with objective at L and eyepiece at S, will see the 
divisions of the scale move through the field of view as the 
mirror turns round. The fii-st method called "the objective 
method," in which the source of light is at /" and the motion 




Fig. 60. 



156 LIGHT XXXIV 

of the image at K is observed, is now very commonly employed. 
The apparatus is compact and the readings can be taken more 
quickly though not so accurately as with the other method. 
The objective method will be generally employed when, as in 
the case of a Wheatstone bridge, we do not wish to measure 
the deflections of a galvanometer mirror accurately, but only 
to work with electrical adjustments such that there should 
be no deflections (null methods). It will also be employed 
when, as for instance in electrometer work, the unavoidable 
sources of error are so large that extreme accuracy in the 
reading of deflections would be waste of labour. 

The second or "subjective method" will generally be used 
when great accuracy is required ; but it is necessary, in order 
to secure this end, that the galvanometer mirrors should not 
be too small and that the whole scale of the dimensions should 
be increased. This of course might also be done with the 
objective method but other difficulties would then arise. The 
objective method requires that the scale should be properly 
shaded so that the spot of light may be easily seen ; the 
subjective method requires that the scale should be properly 
illuminated so that its divisions should clearly appear in the 
telescope. The former condition is more easily secured when 
the apparatus is of small, the latter when it is of large dimen- 
sions. The large dimensions have the additional advantage 
that the observer works sitting at some distance from the 
instrument and hence irregular disturbances due to his moving 
about are often avoided. 

In order to secure accuracy, it is important to adjust the 
position of the scale, and to apply a few corrections to the final 
result. We shall describe the adjustments for the case of in- 
struments in which the subjective method is used, as they are 
then of special importance and may more easily be carried out. 
The student will have no difficulty in applying what is said to 
be the objective method. We shall assume the mirror to turn 
round a vertical axis. The plan of the arrangement is that 
shewn in Fig. 60, but it is clear that the telescope must be 
placed so that its axis passes either above or below the scale. 
The scale, mirror and telescope will therefore all be at different 



XXXIV MEASUREMENT OF ANGLES BY THE OPTICAL METHOD 157 



levels and the telescope must either point downwards or up- 
wards. This introduces no error in the result as long as OX 
in Fig. 59 is taken to be the horizontal projection of the optical 
axis of the telescope, as will be seen by considering that the 
image of each vertical scale division will also be vertical. If 
the telescope is therefore turned round a horizontal axis, the 
same division will alwax's remain against the cross wire, and 
though we may have to shift the scale and mirror in a vertical 
plane in order that the image may appear in the field of view, 
no correction need be applied for want of horizontality of the 
optic axis. If the diameter of the mirror is equal to half that 
of the telescope, we can only obtain the maximum amount of 
light if the optic axis cuts the centre of the mirror, and in any 
case it is best to secure this for reasons of symmetry. The 
adjustment may be made by focussing the telescope on the 
mirror instead of on the image of the scale, moving it if 
necessary until the cross wires cut the mirror symmetrically ; 
a little practice will however enable the observer to make the 
adjustment without altering the focus of the telescope, for its 
field of view is generally sufficiently large to see in addition to 
the scale a blurred image of the galvanometer parts immediately 
surrounding the mirror. If the clear image of the scale is seen 
in the centre of the field of view, the adjustment is sufficiently 
accurate. 

In the equations (1), (2) and (3) it is assumed that the scale 
stands at right angles to the horizontal projection of the optic 
axis, and this adjustment must be made to the necessary degree 
of accuracy. This may etisily be done, as the following investi- 
gation shews that the error intro- 
duced by a slight deviation from 
the correct position can readily be 
eliminated from the result. 

Let the projection of the optic 
axis cut the scale at X (Fig. 61) 
an<l the mirror at 0, also let A and 
B be points on the scale equidistant 
from X, then if the scale is properly 
adjusted OA = OB. 

The scale occupying the position Fig. 61. 




158 LIGHT XXXIV 

A'R, let 7 be the angle BXB' and </> be the angle between OX 
and OB. 

The triangle OXB' gives 

B'X sin<^ sin <i> 

OX ~ . ( IT \ ~~ cos (7 — <^) 



or B'X=^OX -^-^A (4). 

cos 7 4- sin 7 tan <f> 

The angle 7 is supposed to be so small that its cube may be 
neglected compared to unity, and (f> being also small we shall 
neglect the higher powers of 7tan</>; the result will shew when 
these simplifications may safely be made. If then we put 

cos7=l— J7-, sin 7 = 7, 
equation (4) becomes 

B'X= , OXl^ncj > 0Ztan</)(l-7tan<^ + l72) 

1+7 tan <p — J7^ ^ ^ ^ 

approximately. 

If the scale had been placed in its proper position the line 
OB' would have cut it in B so that 

BX = OX tan (/>. 
The error in the reading will therefore be 

OX tan </) (- 7 tan </) + ^7^). 

If the mirror is deflected through the same angle but to 
the other side we find similarly the error of reading 

OX tan (f) (7 tan <!> + ^y^). 
If then 7 is so small that 7^ may be neglected the readings of 
the scale will be as much too great on one side as they are too 
small on the other, and hence if the mean of the two deflections 
is taken the error will only be dependent on the square of the 
angle through which the scale is turned. 

Our adjustments must be made therefore with suflScient 
accuracy to allow ^y'^ to be negligible compared to unity. If 
the scale is one metre long and divided into millimetres, we 
may estimate the tenth part of a division and therefore we may 
if necessary aim at an accuracy of one part in ten thousand. 
The angle 7 for which ^^ is '0001, is about 45', so that the 
inclination of the scale should not exceed this value. The 



XXXIV MEASUREMENT OF ANGLES BY THE OPTICAL METHOD 159 

difference in the deflection right and left are according to the 
above formulae 

2OZ7 tan» (t> = 2X^7 tan <^. 

In actual experiments tan (f> is not likely to be more than J, and 
if 7 is 01, the differences in the reading would be XB/200, 
If the scale is a metre scale so that the greatest value of 
XB is 50 centimetres the difference in the reading of equal 
deflections on the two sides will for the greatest deflection 
which can be read by the instrument be 25 mms. If then the 
mirror is deflected so that the end of the scale appears against 
the cross wire, and two ecjual deflections on opposite sides do 
not give a difference in the reading of more than two and a 
half scale divisions or generally *0025 of the total length of the 
scale, the adjustment is sufficient even if the greatest possible 
accuracy is required. As it is difficult to produce with certainty 
equal deflections, it is advisable to test the position of the scale 
by some direct measurement. 

Fig. 61 shews that 

OA' XA' 

OA' ^ 

hence 



OB' XB' ' 
OA'-OB XA'-XB' 



OB' XBf 

Substituting XA' - XB' = 2X^7 tan 
we find to the required accuracy 

OA' -0B' = yAB, 

which determines the greatest allowable difference between the 
lengths OA' and OB'. 

By means of a string or long rod it is not difficult to secure 
that the two ends of the scale shall be at the same distance from 
the centre of the mirror to within the hundredth part of the 
length of the scale, which will secure a maximum value of 01 
for 7. If the mirror is enclosed in a case with a glass window 
it will be sufficient to measure to the centre of the window. 

To obtain the angle of deflection we use the formula (1) and 
deduce 

2a = arctan ^ y . 



160 LIGHT XXXIV 

For small angles it is often more convenient to calculate 

KX 

arctan yyy than to look it up in a table of tangents. 

Using the series 

arctan </> = </) — J <^^ + -J</>'' — . . . &c., 
we obtain by substitution, writing x for KX and d for OX, 

''~2U 3rf»'*'5rf» '"j'UV Sd'~^5d'~ '")' 

The convenience of the formula consists in the fact that the 
terms after the first may either be neglected or determined 
approximately and that if only relative values are required 
we only need to calculate the bracket. 

It is often necessary to find tan a, sin a, or sin ^ , instead of 

a and the following approximate formulae may then be employed : 

tan<^ = ^[l-gy 



2 U \ 2 VW J • 



sm 



These formulae will be sufficient for nearly all purposes, but 
in case the third term of the series has to be taken into account, 
the first formula becomes 

If it is not necessary to obtain the values of tan (f>, but only 
values which are proportional to them, it is easier to calculate 
the value of 

2c^ tan </>, 2d sin <\> or ^d sin - . 

Thus for instance, if x is 392*3 scale divisions and if the 
distance d of the scale from the mirror is 2500 scale divisions 
the calculation might be made and tabulated as follows : 



XXXIV MEASUREMENT OF ANGLES BY THE OPTICAL METHOD 161 

x= 392-3 

^ = •07846. 

(^J=00616. -.(^y= - 2-41 

(^/= 00048. .{£)'= +^^ 

2d tan 6 = 3901 



In the above investigation it has been assumed that the 
normal to the mirror is at right angles to the axis of rotation, 
and this is sometimes difficult to secure with great accuracy. 
Calculation shews however that when the deviation of the 
normal does not exceed one degree the results are sufficiently 
accurate for nearly all purposes, provided that the distance d is 
measured from the centre of the mirror along the line which 
stands at right angles to the plane of the scale, to which the axis 
of rotation is supposed to be parallel. If as is commonly the 
case the axis of rotation is vertical, d should be the horizontal 
distance between the centre of the mirror and the plane of the 
scale. 

Should it be necessary in exceptional cases to take account 
of the inclination of the mirror, this can be done by adding 
xa (y — a) to the observed deflection x, where a and 7 are the 
angles which a plane drawn at right angles to the axis of rota- 
tion forms with the normal to the mirror and the optical axis 
of the observing telescope respectively. The angles a and 7 
must of course be measured towards the same side of the 
plane *. 

* See F. Kohlrausch, Wiedemann Annalen xxxi. (1887). 



8. P. 11 



SECTION XXXV. 



THE SEXTANT. 



Apparatus required : Sextant, fixed marks. 

The sextant consists of a graduated arc of a circle AB 
(Fig. 62), of about 60°, and a movable arm /, which turns about 




Fig. 62. 

the centre of the arc, and is fitted with a clamp, a tangent screw, 
and a vernier by means of which its position on the graduated 
arc can be accurately determined. 

IG is a plane mirror attached to the arm I, and is called 
the Index Glass. HG is a second mirror fixed to the frame, 
and known as the Horizon Glass. Its upper half is left un- 
silvered. 



XXXV 



THE SEXTANT 



163 



T is a small telescope directed towards the mirror HG 
and placed parallel to the plane of the arc. By means of a 
screw at the back of the instrument, the telescope can be 
moved at right angles to this plane so as to vary the proportion 
of light received from the silvered and unsilvered portions of 
the horizon glass. The horizon glass is so placed that a ray of 
light passing from the centre of the index glass to the centre 
of the horizon glass, is after reflection directed along the axis 
of the telescope (Fig. 63). Let such a ray come originally 
from an object Q, and let another ray coming from a second 




Fig. 63. 



object P, pass through the unsilvered part of the horizon glass 
and proceed in the same direction. The two objects when 
viewed through the telescope are then seen to coincide, one 
being viewed direct and the other after reflection from the two 
mirrors, and the angle between the two mirrors, when this is 
the case, is half the angle between the rays from the objects. 

The graduations on the arc are numbered to read double 
their real value ; hence the reading on the arc gives directly 
the angle between two lines one of which is drawn from the 
centre of the index glass to Q and the other fiom the centre 
of the horizon glass to P. If the distance of the objects is 
sufficiently great this will be sensibly the same as the angular 
distance between P and Q at the observer's eye. 

11—2 



164 LIGHT XXXV 

Adjustments. 

1. The plane of the index glass IG should be at right 
angles to that of the graduated arc. 

Verify that this adjustment is correct by setting the index 
at about 100°, placing the eye near the index glass, and looking 
obliquely at the glass so as to see at the same time part of the 
arc direct, and also its reflection in the glass. If the two 
appear to be in the same plane, the adjustment is correct, and 
the adjusting screws of the index glass need not be altered. 

2. The plane of the horizon glass HG should be at right 
angles to that of the arc. 

Hold the instrument so as to view directly with the telescope 
some small distant object. On turning the index arm round, an 
image of the object, formed by reflection at the two glasses, 
will cross the field. If the two glasses are accurately parallel, 
this image can be made to coincide exactly with the object 
seen direct. If the plane of the horizon glass is not at right 
angles to that of the arc, so that the two mirrors are not 
parallel, the image will appear to pass on one side or other of 
the object. 

By adjusting the top screw at the back of the horizon glass, 
the image seen after two reflections, and the object seen directly, 
can be made to coincide exactly in one position of the index 
arm. 

When this is the case the two mirrors are parallel, and the 
horizon glass is at right angles to the plane of the arc. 

3. The axis of the telescope should be parallel to the plane 
of the graduated arc. 

To test whether this condition is fulfilled with sufficient 
accuracy, place two small sights of exactly equal height on 
the divided circle. These sights are conveniently made by 
bending strips of sheet brass into two parts at right angles 
to each other, so that when resting on the sextant their upper 
edges are horizontal and at the same level as the axis of the 
telescope, the sextant being placed on a horizontal table. If 
the sights are placed as far apart from each other as possible 
and in suitable positions, the same distant object may either 
be viewed through the telescope or by the naked eye looking 



XXXV THE SEXTANT 165 

over the sights. A point on the distant object which may be 
brought into contact with the upper edges of the sights when 
looked at by the naked eye, should then appear in the centre of 
the field of view. For accurate work with the sextant the ad- 
justment of the telescope should be con*ect to within 10 minutes 
of arc or '003 in angular measure. Hence if the sights are 
placed at a distance of 15 cms. from each other, they should 
be carefully constructed so that the two upper edges are hori- 
zontal and within half a millimetre of the same height above 
the divided circle. If the adjustment is found wrong it may 
be corrected in well-made sextants by two small screws in the 
frame which carries the telescope. 

Determination of the Index Correction. 

When the two mirrors are parallel, it may be found that 
the Vernier index does not read exactly zero. The reading is 
termed the index error, and the index correction is the index 
error with the sign reversed. 

To determine the index correction, direct the telescope to a 
distant object and turn the index arm till the two images of 
the object appear in the field of view. Then clamp the index 
arm, and bring the images into coincidence by means of the 
tangent screw. Read the Vernier, and notice whether the 
reading is positive or negative. 

The reading of the Vernier with the sign reversed is the 
index coiToction. 

Exercise. 

Standing on the given spot, determine the angle subtended 
by the two given points. 

Hold the sextant in the right hand, with the arc downwards. 
Look through the telescope at the lower of the two objects, 
holding the sextant so that tlie plane of the arc passes through 
both objects. Move the index arm along the arc till both 
objects appear at the .same time in the field of view, then clamp 
the arm by means of the screw behind the scale, and bring the 
two images into coincidence by means of the tangent screw. 



166 LIGHT XXXV 

Read the position of the index, repeat the observation, and 
record as follows. 

10 May, 1899. 

The reading when the two images of the same distant object 
coincided was found to be as a mean of 3 observations — 2' 12". 
Readings when the images of the two given points coin- 
cided : 

11° 30' 20" 
30' 30" 
30' 00" 



Mean IV 30' 17' 

Index Correction + 2' 12' 



. •. Angle subtended by the given points] ^ , „ 

at the point of observation J 

It will often be found impossible to find suitable objects at 
a great distance. If the distance is less than 1000 yards, and 
the measurements of different observers are to be compared 
with each other, care must be taken that the observer does 
not change his position during the observations and that the 
position is well marked. Students will get a good idea of the 
delicacy of the observations by observing two objects at a 
distance of 1000 yards and at an angle of not less than 10° 
from each other; receding from or approaching the objects 
by a yard ought to make an appreciable difference in the 
coincidence of the images. 

It would be difficult to name a more useful or instructive 
instrument than the sextant, and, if time allows, students should 
obtain a little practice in the determination of latitude and 
local time by means of it. The study of errors introduced by 
imperfections in the adjustment of the different parts, will form 
an excellent foundation for the study of other optical instru- 
ments. 

Consult: Chauvenet, Spherical and Practical Astronomy, 
vol. II. 



SECTION XXXVI. 

CURVATURES AND POWERS OF LENSES. 

Apparatus required : T^uo large lenses, one concave, one 
cojive./', one suiall convex lens, spherometer, scale, plane mirror, 
and simple optic bench. 

The radius of curvature of a spherical surface may be found 
by means of a spherometer, or by using the surface as a mirror, 
and determining the positions of conjugate points. Both 
methods will be used in what follows. 

Place the spherometer provided on a plane surface, e.g., a 
sheet of glass, and turn the milled head till the point of the 
screw just touches the surface. The exact point may be known 
either by the slight increase of the resistance to rotation of the 
screw, or by the rocking of the spherometer on the surface as 
soon as the point of contact is passed. 

Take three readings of the zero of the spherometer in this 
way, and then place it on the surface the curvature of which is 
required, in this case a lens. 

Again turn the screw till the point just touches the surface, 
making the adjustment three times and 
taking a reading each time. 

The pitch of the screw is marked on 
the spherometer. 

In order to determine the radius of 
curvature of the surface from these 
observations, we require to know the 
distances between the feet of the sphero- 
meter. To measure these distances Fig. 64. 
place the spherometer on a sheet of paper, and press it down 
gently, so that the positions of the feet are indicated by slight 




168 LIGHT XXXVI 

depressions in the paper. If these points be A^, A^, A3 (Fig. 64), 
measure by means of a glass scale laid on the paper with the 
scale downwards, or by means of compasses, the distances 
-42^3, A3A1, A^A^, call them a, b, c. The radius of the circle 
CA, circumscribing the triangle ^1^2-^2 will be given by 

abc 
r — 



where 



4VS(,S- 


a){S- 


■b)(S- 


^y 


8 = "- 


2 


c 




•educes to 








r = 


a 

V3- 







In practice a, b, c will not be exactly equal but nearly so 
and in that case we may generally substitute for a the average 
value of the distance in the last equation. 

In order to see within what limits of accuracy this is allow- 
able we write s for the arithmetical mean of a, b, c and put 

a = s + a, b = s + ^, = 5 + 7, 

and by addition 

a + 6 + c = 35+a + /3 + 7, 

or since by definition 

3s = a + 6 + c, 
it follows that 

a + /3 + 7 = 0. 
We may now put 

\/35 (s - 2a) (s - W) {s - 27) * 
If we expand the products we find that there is no term 
containing the first power of a, ^, 7 because these quantities 
only occur in the combination ot + y8 + 7, which is zero. If 
a, /9, 7 are considered as small quantities the cubes of which 
can be neglected and if we write 

«2 = a/3 + /S7 + 07 
= -i(a^ + ;^^ + 7^), 
it is easily found that 



XXXVI 



CURVATURES AND POWERS OF LENSES 



169 



As an accuracy of one part in 500 will probably not be 
reached in this exercise unless the mean of a large number of 
determinations are taken, we may write 



r = 



whenever - is less than 
s 



25 



V3' 

If the second term cannot be 



neglected it is easily computed with sufficient accuracy. 

Now consider a section of the surface by a plane through 
AC and the centre of curvature of the surface 0, i.e., the 
circle ANDN' (Fig. 65). The plane 
through the points ^1^0-^3 is represented 
by the line AGD. 

Assuming that the spherometer is 
properly constructed so that when the in- 
strument is placed on a horizontal surface 
the point of the screw touches the surface 
at the centre of the circle circumscribing 
AxA<iAi, the distance {d) through which 
the screw has been moved will be CN 
when the instrument is placed on the spherical surface. 
Now GN.CN'^CA^ 

and CN' :=2R-CN where R^OA. 




Fig. 65. 



Hence 



or 



d{2R-d) = r\ 
J, 1 r' + d^ 



Calculate R from this equation and determine similarly the 
radius of curvature of the other surface of the lens. 

If the point of the screw instead of touching C passes 
through a point at a distance h from it, the error in the measured 
distance will be — h*/'2R, and the fractional error of the radius of 
curvature measured will be It^/r^. 

Verify by the following optical method the values found : 

I. For a Concave surface. 

Place the screen provided with the hole and cross wires at 
one end of the optic bench with the white side towards the 
centre of the bench. Place a bright Hame behind the screen. 



170 



LIGHT 



XXXVI 



Place the lens on the bench with its axis parallel to the 
axis of the bench, and the surface the curvature of which is 
to be measured towards the screen. Find the position of the 
surface when the image of the cross wires on the screen pro- 
duced by reflection at the surface is most distinct. The cross 
wires are then at the centre of curvature of the surface. 

II. If the surface is convex the radius of curvature may 
be determined optically by the following method : — 

C (Fig. QQ) is the lens the radii of curvature of which we 
have to determine. B is an auxiliary lens provided with a stop 




Fig. 66. 

of small diameter. A is the screen in the centre of which is 
a small hole with a cross wire, which is illuminated as shewn. 
Keeping the distance AB, which must be greater than the focal 
length of B, constant, we can find a position of C such that 
the rays of light, after passing through B, strike the surface of 
C perpendicularly, and returning along the same path, form 
an image of the cross wire on A. Take the reading of the 
position of G, then remove G, and place a screen D in such a 
position that a distinct image of the cross wire is obtained on 
it, the lens B being kept in the same position. The distance 
from the lens to D is then the radius of curvature of the 
face nearer the cross wire. By turning the lens C round, the 
radius of curvature of the other face is similarly obtained. 
The apparatus used is shewn in Fig. 67. 




Fig. 67. 



XXXVI CURVATURES AND POWERS OF LENSES 17 1 

Deterraine approximately the focal length of the lens, if 
convex, by placing a screen behind it at such a distance that an 
inverted image of some distant object is formed on the screen. 
The distance of the screen from the nearest point on the surface 
of the leus is the focal length approximately. 

To determine it more accurately, if the lens is a converging 
one, mount it with a plane mirror immediately behind it. 
Place the mounted lens and the mirror on the optic bench. 

Behind the hole and cross wire place a luminous burner, 
and find the position of the lens and mirror for which a distinct 
image of the cross wire is projected on to the screen, in the same 
plane and as nearly as possible coincident with the cross wires 
themselves (Fig. 68). 




The light from F in the direction FA is refracted by the 
lens and proceeds after refraction in the direction AD. li AD 
is perpendicular to the surface of the mirror, the reflected ray 
traverses the same path as the incident, and is, therefore, 
brought to a focus again at F. 

In order to determine the distance of the surface of the lens 
from the screen read the position of some part of the lens-stand 
on the scale of the optic bench used. Move the lens towards 
the screen till it touches one end of a rod of known length the 
other end of which touches the cross wire. Again read the 
position of the lens-stand. The sum of the length of the rod 
and the distance through which the stand has been moved is 
the distance of the surface of the lens from the cross wire. The 
focal length may be found from this by adding to it J the thick- 
ness of lens. 

If the lens is a divergent one, place between it and the cross 
wires a convergent lens. Place behind the two and close to 



172 



LIGHT 



XXXVI 



the divergent lens, a plane mirror (Fig. 69). Move the diver- 
gent lens and plane mirror till a distinct image of the cross 
wires is formed on the screen close to the wires. Note the 
position of the lens, remove it and place behind the converging 
lens a screen in such a position that an image of the cross wires 




-^*-E- 



Fig. 69. 

is formed on it. The distance from the screen to the position 
previously occupied by the lens is the focal length required. 

The focal length of a lens is connected with the radii of 
curvature of its surfaces and with the index of refraction /i of its 
material by the equation 



7=('*-^>(i+ii 



where R^ and i^ are taken positive for convex and negative for 
concave surfaces. 

From this equation we get 



flKR.'^R)' 



Calculate /x from the observations, and tabulate as follows : 
10 February, 1899. 
Lens No. 2 double convex. 

a = — '03 cms. t- — '05 

^ = -•14 „ ^V«^ = -0014 

7 = + -!'^ .. 
„ and r = 3*560 „ 

di by spherometer = '1564 cm. 



a = 6-14 
b = 6-03 
c = 6-34 
5 = 6167 



cms. 



Hence 



By reflection 



Hence 



0/2 „ 

f (measured) 
fj, (calculated) 



= -1567 „ 

= 40*60 cms. 

= 40-52 „ 

= 39-9 „ 

= 40-0 „ 

= 40-5 „ 
= 1-50 



SECTION XXXVII. 

DETERMINATION OF THE INDEX OF REFRACTION 
OF A LIQUID BY TOTAL REFLECTION. 

Apparatus required : Horizontal graduated scale with 
upright and dit, ebonite block, and glass cube. 

When a ray of light traversing an optically dense medium 
impinges on the surface of separation of that medium from 
a rarer medium, making an angle with the normal at the point 
of incidence greater than the " Critical Angle," the ray is totally 
reflected, no part of it entering the rarer medium. The least 
value of the angle of incidence for which total reflection can 
take place, i.e., the Critical Angle, is, if N is the index of 
refraction of the denser, n that of the rarer medium, given by 
the equation 

sin 6 = njN. 

A glass cube of about 4 cms. edge is provided. On one face 
a line parallel to and about a cm. from an edge has been drawn 
with a diamond. 

Put a few drops of water on the ebonite block provided, and 
place on it the cube with the face on which the line is drawn 
vertical, the line itself horizontal, and a cm. above the ebonite. 
A film of water will be formed in immediate contact with the 
lower face of the cube. Now place the ebonite and cube on 
the scale and about 30 cms. behind the slit. Place the scale 
horizontally on a table in front of a window through which the 
sky can be seen. Look through the slit "at the lower surface of 
the cube, and notice that on moving the cube and block towards 
the slit, the appearance of this surface changes from bright to 



174 



LIGHT 



XXXVII 



dull, and at the point of change a coloured band extends across 
the surface from left to right. Adjust the distance of the cube 





Fig. 70. 

from the slit till that part of the coloured band where violet 
shades off into green coincides with the diamond scratch on the 
face of the cube. 

Measure h^ the height of the slit, and h^ the height of the 
line on the cube, above the surface of the scale, and d the 
horizontal distance between the face of the cube and a per- 
pendicular let fall on to the scale from the slit. 




Fig. 71. 



XXXVII 



TOTAL INTERNAL REFLECTION 



175 



Then if i = angle of emergence of the ray from the cube, 
(Fig. 71), 

tan I — -1 — . 

If r = the angle of incidence of the internal ray at C, and 
N = the index of refraction of the cube, then 

sin 1 = iV sin ?*. 

Since total reflection just begins at B, i.e., the angle of 
incidence at 5 is the critical angle, 

71 = N cos 7', 

where ?i = the index of refraction of the liquid at B. 
Squaring and adding the last two equations we get 
?i2 + sin^ i = N^ 
or n = ViV^ — sin' i. 

From which n may be calculated if N is known and i is found 
by measurement of /i,, ho, and d. 

Determine by this method the value of N for the cube, 
taking the index n for water to be 1'333. 

Having found N find n for ethyl alcohol. 

Arrange your observations as follows : — 

25 January, 1893. 
Water film h^ = 21*5 cms. d = 2l'S cms. 

A,= 31 „ 

18*4 
tan i = 2^3 =862 

.-.1 = 40" 45', sin I =653 
N' = (1-333)' + (653)2 = 22060 
.•.iV^= 1-485 



Alcohol film 



A, = 21*5 cms. 
^= 31 „ 

^--• = 2l| = -^21 

1 = 35° 48', sin I =586 
7i» = (1-485 )«-(-586)« 
.-. 7i = 1-365. 



d = 25*5 cms. 



SECTION XXXVIII. 

MAGNIFYING POWERS OF INSTRUMENTS. 



Apparatus required : Telescope, microscope, millimetre 
scales. 

When an object is viewed through an optical instrument, 
such as a telescope or microscope, the apparent size of the 
image is in general larger than that of the object as seen 
direct, and it is the purpose of this exercise to determine the 
amount of magnification. 

The magnifying power of a telescope is defined as the ratio 
of the apparent size of the object as seen through the telescope 
to the apparent size of the object as seen direct. 

To determine this ratio the telescope is directed towards 
some surface on which is a regular well-marked pattern of some 
kind, such as the dividing lines in a brick wall, and the pattern 
looked at with one eye through the telescope, while with the 
other eye the pattern is seen direct. Count how^ many times 
some convenient length in the magnified pattern contains the 
same length of the pattern seen direct. This number is the 
magnifying power of the telescope as used. 

Alter the distance of the telescope from the pattern, and 
again find the magnifying power, which should be the same 
as before if the distances are great compared to the focal 
length of the object lens of the telescope. 

The magnifying power of a microscope is defined as the 
ratio of the apparent size of an object as seen through the 



XXXVIII MAGNIFYING POWERS OF INSTRUMENTS 177 

instrument, to the apparent size of the object when placed at 
the nearest distance of distinct vision and viewed direct. 

Place a millimetre scale on the stage of the microscope, and 
focus it distinctly. Hold a paper scale graduated in millimetres 
about 20 cms. below the eye and parallel to the scale on the 
stage, move it up and down till it is as near the eye as is con- 
sistent with distinct vision. Use one eye to view the scale on 
the stage through the microscope, the other to view the paper 
scale direct, and note how many divisions of the paper scale 
correspond to one of the stage scale. This is the magnifying 
power required. 

Describe your experiment in your note-book and enter the 
results as in the previous exercises. 



8. p. 12 




SECTION XXXIX. 

ADJUSTMENTS AND USE OF THE SPECTROSCOPE. 

Apparatus required: Spectroscope, platinum wires for 
heads, salts, and crayons. 

Vision through a prism. Let a luminous point S (Fig. 72) 
send out a pencil of homogeneous 
light, the rays of which, after re- 
fraction through the prism, seem to 
diverge approximately from a point 
;S'. An eye at E will therefore see 
the luminous point in a displaced Fig. 72. 

position, the amount of displacement 

depending on the refractive index of the prism for the rays. If 
the light sent out by the luminous point is not homogeneous, 
but is of two kinds having different refractive indices, there 
will be two separate images side by side, which will be coloured 
differently if the difference in the refractive index is sufficiently 
great. If, finally, the luminous point sends out white light, there 
will be an infinite number of images shewing a succession of all 
the colours of the spectrum from red to violet. 

Seal about four or five centimetres of platinum wire of 
*2 mm. diameter into a piece of glass tubing, and bend the end 
of the wire into the shape of a loop thus : 

Fig. 73. 



XXXIX SPECTROSCOPE ADJUSTMENTS 179 

Wet the loop slightly, dip it into a mixture of common salt and 
borax in equal proportions, and heat carefully in the edge of 
the flame of a Bunsen burner, until the salt fuses and forms 
a transparent bead. If the loop is not completely filled by the 
bead, place some more of the substances on it and repeat the 
process of fusion. Then fix the glass tube to the stand pro- 
vided, and place it so that the bead touches the flame and 
colours it. The platinum wire should dip slightly downwards 
(Fig. 73), otherwise the bead has a tendency to run back along it. 

Hold the loose prism provided in front of your eye, and turn 
your head until you see the displaced image of the flame. 
Next take a second piece of wire, and having prepared a bead 
of a lithium salt, place it in the flame simultaneously with the 
sodium bead. Make a sketch in your note-book of what you 
see. What conclusions do you draw respecting the light sent 
out by a flame containing both sodium and lithium ? 

Look now at a luminous flame, again sketch the appearance 
in your note-book, and explain it. 

Cut a slit not broader than a millimetre in a piece of card- 
board. Hold the cardboard in your left hand so that the slit 
is in front of a Bunsen burner containing both a lithium and 
sodium salt, hold the prism in the right hand in front of your 
eye and look at the image of the slit. Sketch the appearance 
in your note -book. 

Such a combination of slit and prism forms the simplest 
kind of spectroscope, and is often useful for the rapid examina- 
tion of light sent out by a flame, or an electric discharge. 

In practice the virtual image of the slit is magnified by 
being looked at through a telescope. But this cannot be done 
with advantage without some further change in the optical 




Fig. 74. 

arrangement, to obviate the so-called " aberration " of the rays. 
For if rays of light diverging from a point are traced through a 

12—2 



180 LIGHT XXXIX 

prism, it is found that after emergence they do not accurately 
diverge from a point, but the section of the pencil will have 
the appearance of Fig. 74. This does not sensibly affect the 
sharpness of the image when looked at with the naked eye, or 
even under small enlargements such as are used in pocket 
spectroscopes, but when higher power is required the aberra- 
tion must be obviated. This may be done by the introduction 
of a lens placed in such a position between the slit and the 
prism, that the beam of light coming from any point of 
the slit becomes parallel before it falls on the prism. As a 
parallel pencil of light will remain parallel after refraction at 
any number of plane surfaces, there is no aberration in this 
case. We thus arrive at the arrangement shewn in Fig. 75. 




Fig. 75. 

A tube carries at one end the slit S and at the other the lens L, 
the slit being in the focal plane of the lens. This tube is called 
a "collimator." The pencils of light rendered parallel by L 
fall on the prism P, and after passing through it are received 
by a telescope T. The eye, on looking through the telescope, 
will see an image of the slit, which is displaced by the prism, 
but is sharp provided the slit is illuminated by homogeneous 
light. If, on the other hand, the light falling on the slit consists 
of groups of waves differing in wave-length, each group will 
give a separate image of the slit. If further the wave-lengths, 
and therefore the refrangibilities, vary continuously over a 
certain range, the images of the slit will lie side by side or even 
overlap, so that a continuous band of light will be seen. 

We call the appearance presented when the light of a 
luminous body is examined by means of a spectroscope the 
"spectrum" of the body. We say that the body has a 



XXXIX SPECTROSCOPE ADJUSTMENTS 181 

"contiDuous spectrum" when the band of coloured light is 
continuous. We say, on the other hand, that the body has 
a "line spectrum," if a number of separate coloured lines are 
seen, which students must be careful to remember are only 
images of the slit. If the slit is curved the lines will be 
curved, if the slit is broad the lines appear broad, and nan-ow- 
ing the slit will narrow the lines down to a certain limit, which 
depends on the diameter of the object-glass used. A "band 
spectrum " is a spectrum consisting of bands which are broad 
even with a narrow slit. These bands are often sharp on one 
side and fade away gradually on the other. 

As the spectrum of a body, whether it is a line or band 
spectrum, is found to be characteristic of the body, it is neces- 
sary to determine the positions of the lines and bands on some 
convenient scale. 

The arrangement generally adopted in one-prism spectro- 
scopes is shewn in Fig. 76. A small tube 
MQ has at one end a scale of fine equi- 
distant lines Q, at the other end a lens M, 
the scale being at the principal focal plane 
of the lens. The tube is placed so that the 
light rendered parallel by the lens M, is re- 
flected at the surface of the prism into the 
telescope T. When the scale is illuminated by 
a small gas flame, the observer sees not only 
the spectrum of the body, but superposed on 
it the image of the divided scale, and he can ^*^' ^^' 

read off the position of each separate line on this scale. 

Fig. 77 shews a spectroscope consisting of collimator, prism, 
telescope, and scale- tube. The collimator is provided with a 
projecting metal sheet, and the scale-tube with a projecting 
metal cylinder, to prevent the flames used being brought so 
near as to injure the instrument. 

Adjustments of the Spectroscope : It has been shewn 
above that the light leaving the collimator should consist of 
parallel rays. To secure this, if the distance between the slit 
and collimator lens can be altered, the telescope and collimator 
are adjusted as follows : — 




182 



LIGHT 



XXXIX 



The telescope is taken off the spectroscope stand, and a 
distant object is observed through it. If the object is seen 




Fig. 77. 

clearly, the telescope is in adjustment for parallel light. If not 
it should be adjusted and then replaced. Illuminate the slit of 
the collimator by a sodium flame, and move the collimator in 
and out till the image of the slit seen in the telescope is sharp 
and distinct. Place a small luminous burner in front of, but 
not too near to, the end of the scale-tube, and adjust the scale- 
tube till both the scale and the sodium line are. seen sharp 
at the same time. Move the scale-tube at the same time until 
the sodium line stands at a convenient number of the scale*. 
The best test of good adjustment is the absence of parallax; 
that is to say when the eye is moved a little to the left or right 
the sodium line should not change its position relative to the 
scale. Accurate measurements with the spectroscope are impos- 
sible when parallax exists to an appreciable degree. 

If the position of the slit of the collimator cannot be changed, 
the telescope must be adjusted by illuminating the slit with a 
sodium flame, and focussing the telescope so that the yellow line 
appears sharp. The scale is then focussed as described. 

Before proceeding with any further measurements, consult 
one of the Demonstrators to see that your adjustment is correct. 
With a narrow slit the sodium line should just appear double. 

As the question of brightness of spectra is often an im- 

* As the scale divisions may differ in different instruments, a label is 
attached to each spectroscope, on which the number with which the sodium 
line should be made to coincide is indicated. 



XXXIX SPECTROSCOPE ADJUSTMENTS 183 

portant one, it will be useful to remember the following facts. 
Let SL (Fig. 78) be a horizontal section of the collimator, S 
being the slit and LL the lens. Let AB be a flame sending 
out light the spectrum of which is to be examined. The 
light from the part of the flame near A will illuminate the 
part of the collimator lens near Z, the rays of light going from 
A through the slit to Z, similarly the light from the part 
of the flame near B will illuminate the part of the lens 
near L. The whole collimator lens is thus filled with light. 
When the whole collimator levs is filled with light maximum 
brightness of the spectrum is secured. 




Fig. 78. 

If the burner is placed at ^^^j close to the slit, it is only 
the light from the central parts of the flame which reaches the 
collimator lens : the rest of the light falls on the inside walls of 
the tube, which are blackened to prevent reflections. In this 
case, therefore, although the flame is nearer, a smaller part of 
the flame illuminates the collimator lens, so that nothing is 
gained in brilliancy, the brightness of the spectrum being 
exactly the same as before. If, on the other hand, the flame 
is too far away at A^Bo, the outer portions of the lens are not 
illuminated at all ; in that case the brightness of the spectrum 
is reduced. When the collimator lens is not filled with light, 
as in this last case, beside loss of light there is the disadvantage 
of worse definition, so that in spectroscopic work it is generally 
of importance that the whole collimator lens should be illumi- 
nated, but when that is secured no further increase in brightness 
is possible. 

Where the source of light is very small, as in the case of an 
electric spark, with a narrow slit the collimator lens would not 
be filled with light unless the source were brought inconveni- 
ently near the slit. A lens P (Fig. 79), should therefore be 
placed in such a position between the spark E and the slit 8 
that the rays traverse the paths shewn, and an image of the 



184 LIGHT XXXIX 

spark is formed on the slit. Here, again, it does not matter 
how the distances of lens and spark from the slit are adjusted, 
nor how bright is the image of the spark seen on the slit plate. 
The filling of the collimator lens with light is the sole circum- 
stance on which the brightness of the spectrum depends. We 




Fig. 79. 

may even move the lens so that the image of the slit is out of 
focus without producing great loss of light. The use of the lens 
has the further advantage of enabling us to examine the spectra 
of different parts of the spark separately. 

Mapping Spectra : When the spectroscope is in adjust- 
ment and the sodium line at the proper scale division, place 
a luminous burner behind the slit of the collimator so as to 
produce a continuous spectrum. Draw the spectrum which you 
see, representing as nearly as possible with the crayons provided, 
the colours and shades and their extent on the spectroscope 
scale. Use for this purpose a sheet of paper on which scales 
with the distance between the scale divisions approximately 
equal to one millimetre are printed. 

Next prepare six pieces of platinum wire sealed into glass 
tubes, as described at the beginning of the section. The platinum 
should not be thicker than '2 mm.; it must be perfectly clean, 
and before use be heated to a red heat in the Bunsen flame. 

Adjust the height of the burner so that the hottest part of 
the flame, which is just above the top of the inner cone, lies 
about the level of the lower portion of the slit. 

Place a small quantity of barium chloride on the platinum 
wire, wetting the latter first with a little dilute hydrochloric 
acid. Support the wire in the stand provided and insert it 
into the hottest part of the flame. After a few seconds the 
upper part of the flame will be green coloured, and on looking 
through the spectroscope you should see a spectrum of bands, 
with a line in the green which is sharp and bright. If this line 



XXXIX MAPPING SPECTRA 186 

does not appear, the adjustment of your spectroscope is probably 
faulty or your slit is too wide. If the spectrum is very faint, 
either the flame is not placed properly in front of the slit, or the 
latter is too narrow. The best width of slit is only obtained by 
trial. For the observation of the spectra of the alkali metals 
with small spectroscopes, the slit ought not to be so narrow 
that the sodium line appears double, but it ought to appear 
a sharp line and not a broad band. 

Represent on the paper scales provided, by meaus of shading 
with a sharp black pencil, the appeai*ance of the bands and lines 
seen in the spectroscope. After you have made an independent 
drawing, consult some atlas of spectra and compare yours with 
the drawing given there. If you tind that you have not seen 
all the bands shown there, repeat your experiment. Some of 
the fainter bands may however escape your notice, as they 
require a completely darkened room. 

Make similar drawings of the spectra of lithium, thallium, 
strontium, and calcium chlorides. Notice that the spectrum of 
the last-mentioned salt gi-adually changes. The spectrum first 
seen is due to the chloride of calcium, and is gradually replaced 
by that of the oxide. Make separate drawings of the first and 
last stages through w^hich the spectrum passes. 

During the observations, check from time to time the 
position of the sodium line. If you find that a slight change 
has taken place, it is not necessary to alter the position of the 
scale-tube, apply instead a correction to your readings. Thus, if 
the sodium line reads 8*1 instead of 8, you may correct all other 
readings by subtracting '1. 

As a last example take a bead of potassium chloride. You 
should have no difficulty in seeing a line in the red, which is 
really a double line. But there is also a line at the extreme end 
of the visible violet, which requires special precaution to see. 
Not to have your eye disturbed by extraneous light, move the 
telescope until the red and green are out of the field of view, so 
that when looking at the continuous spectrum of a gas flame 
you only see the violet ; also remove the burner which illumi- 
nates the scale. Carefully introduce the potassium salt into 
the hottest part of the flame while you are looking through the 



186 LIGHT XXXIX 

telescope ; -svith a little practice this is easily done. If you fail 
to see the potassium line, slightly open the slit. As soon as 
you can see it, illuminate the scale a little and take a reading. 
Finally, move the telescope back, and without altering the 
width of the slit, take a reading of the sodium line. The 
measurement of the potassium lines should be made several 
times in order to secure greater accuracy. 

It will be noticed that while thallium, sodium, potassium, 
and lithium give line spectra, the spectra of the alkaline earths 
consist of bands. These latter are due to the metallic oxides 
or chlorides, while in the case of the alkali metals the salts are 
decomposed and the spectra of the metals themselves appear. 

Instead of making shaded drawings of the appearance of 
spectra, they may be represented diagrammatically, as in 
Fig. 80, where the intensities are represented by the lengths of 
the lines. Thus 5 is a sharp line, 
the intensity of which is estimated 
to be about J that of J., i) a band 
having a sharp edge at 11*30 but 
fading away gradually to the other "pig, 80. 

side, the limit of visibility being 

about 11-7, G, on the other hand, a band which is brightest in 
the middle and ends abruptly at 13-6 and 13*84. 



Iiiiilmilii| i l|inlii'iliiiiliii!liri 



SECTION XL. 

REDUCTION OF SPECTROSCOPIC MEASUREMENTS TO 
AN ABSOLUTE SCALE. 

When the spectrum of a given substance is mapped, the 
relative positions of the lines will be found to differ according 
to the instrument used. Even if the sodium line is at the 
same scale division in each, the readings of the other lines will 
differ in the different instruments, since they depend on the 
dispersion of the prism and on the distance between the divi- 
sions of the scale. In order that measurements taken with 
different instruments may be comparable, it is necessary to 
have some method of representation which is independent of 
the instrument employed. 

It is usual to take the wave-length as that which character- 
ises a given light vibration, and the spectrum mapped according 
to the wave-length scale is called a ** normal " spectrum. In- 
stead of the wave-length scale, we might adopt one which takes 
the " time of vibration " as the characteristic property of a set 
of waves, and such a scale would present many advantages. 
The time of vibration t is connected with the wave-length X 
by the relation X = t;f, where v is the velocity of light. If a 
set of waves in air, from a source sending out homogeneous 
light, enters a different medium, as for instance a glass prism, 
the time of vibration will be the same in the glass as in air, 
but the velocity of propagation of the waves, and hence the 
wave-length, will be different. When a particular kind of 
homogeneous light is defined by its wave-length, we must 
therefore state the medium in which that length is measured. 



188 LIGHT . XL 

The most scientific proceeding would be to take the wave- 
length in vacuo as the standard, but as all direct measurements 
of wave-lengths are taken in air, and the reduction to vacuo 
involves an accurate knowledge of the refractive index of air, it 
is usual to take the wave-lengths in air as standards. To be 
strictly accurate we must define the pressure and the tempera- 
ture of the air, since its refractive index depends on its density. 
Rowland's table of standard wave-lengths is now generally 
adopted, and holds for a pressure of 76 cms. and a temperature 
of 20° C. In certain investigations it is necessary to know the 
wave-lengths in vacuo, and Tables have been constructed 
( Watts's Index of Spectra, Appendix E) by means of which the 
conversion to vacuo may be easily effected. 

In order to avoid having the decimal point at an incon- 
venient place, wave-lengths are usually given not in centimetres 
but in so-called tenth-metres (written X"' metre) the lO^^'th part 
of a metre or the lOHh part of a centimetre. According to this 
scale the wave-lengths of the two sodium lines in air are 
589018 and 5896"15, and the correction to vacuo is + 1*60. 
Spectroscopes containing one prism of the usual size only 
barely separate the sodium lines, and measurements with such 
instruments can only be accurate to two or three X*^'^ metres. 
All figures beyond the decimal point may therefore be left out 
of account in what follows. 

It is sometimes advantageous to consider the " wave- 
numbers," or namher of waves per centimetre rather than the 
wave-lengths. If there are v waves in one centimetre z^A,= l, 
and hence the wave-number is simply the reciprocal of the 
wave-length. For the least refrangible sodium line the wave- 
number is therefore 

Different methods may be used to reduce measurements 
taken on an arbitrary scale to wave-lengths, and it is necessary 
to gain practical experience of the most important of these. 
In each method the positions of two or more lines the wave- 
lengths of which are known, are observed on the arbitrary scale 
of the spectroscope, and the unknown wave-lengths of other 



wave 



XL REDUCTION OF SPECTROSCOPIC MEASUREMENTS 189 

lines obtained from them by interpolation. It is in the process 
of interpolation that the methods differ. 

Method I. Graphical inte-rpolation by means of a wave- 
length curve. 

When the same spectroscope is always used by the same 
observer and its adjustments remain unaltered, it is generally 
simplest to draw once for all a curve connecting the scale read- 
ings with the wave-lengths. To do this, the positions of a 
certain number of so-called "standard lines" are observed on 
the spectroscope scale, and a curve having the scale divisions as 
abscissae and the wave-lengths as ordinates is drawn. The 
shape of the curve will resemble that 
shown in Fig. 81. It is important to 
choose the scale of the curve suitably. 
If the paper is divided into millimetres, 
each scale division may be made to cor- ^^^ 
respond to one millimetre, as it will 
enable subdivisions to be estimated to 
the same degree of accuracy on the curve 
as in observing a line. As regards the 
scale of ordinates, it must be remembered that the visible 
spectrum ranges from 7,900 to 3,900 X'** metres, and that it 
will not be possible to determine with an ordinary one-prism 
spectroscope the last figure accurately. Hence if one millimetre 
is made to correspond to 20 X'** metres, the scale will be suffici- 
ently large, and the total length of paper necessary for the curve 
will be 20 centimetres. 

Once the curve is drawn the wave-length of any unknown 
line may be obtained immediately by inspection. A list of 
convenient standard lines is given at the end of this section. 

Exercise I. Draw a curve connecting the wave- 
lengths with the scale readings of a spectroscope. 

From the observations you have made of the lines whose 
wave-lengths are given in the Table at the end of the section, 
draw the curve a« described, and prepare a Table giving the 
wave-length for each tenth scale division of the instrument you 
have used, thus : 



Scale reading 

Fig. 81. 



190 



LIGHT 



XL 



Spectroscope A. 
Sodium lines placed at 8. 



Scale Division. 


Wave-Lengths in 
Xth metres. 


4 


7862 


5 


7239 


6 


6669 


7 


6281 


8 


5893 


9 


5622 


10 


5351 


11 


5180 


&c. 


&c. 



The Table should extend over the whole of the visible 
spectrum. 

Method II. Graphical Interpolation by means of a curve 
of wave-numbers. 

The curve given in Fig. 81 has a some- 
what large curvature, hence the number of 
standard lines required to draw it accu- 
rately is comparatively large. To obtain a 
curve w^hich is more nearly straight, and 
can therefore be drawn with greater accu- 
racy, we make use of the wave-numbers v as Fig. 82. 
ordinates instead of wave-lengths X (Fig. 82). 

To find the wave-length of an unknown line its scale reading 
is taken and the corresponding value of v found by the curve. 
The required wave-length will then be the reciprocal of the 
value found. The wave-numbers per centimetre of the red 
and the blue potassium lines are respectively 13016 and 24717. 
If therefore, each millimetre of the squared paper corresponds 
to 50, the vertical distance required will be 24 cms. 

Exercise II. Draw a curve connecting the inverse 
wave-lengths or wave-numbers with the scale readings. 

Prepare a second Table from this curve, thus : — 



XL 



REDUCTION OF SPECTROSCOPIC MEASUREMENTS 
Spectroscope A. Sodium lines at 8. 



191 



Scale Division. 


Wave-Length. 
cms. 


Wave-Numbers. 


4 
5 
6 
7 
8 


7-860 X lO-** 
7-238 
6-712 
6-277 
5-893 
(fee. 


12723 
13816 
14895 
15931 
16969 
<fec. 



Method III. Graphical interpolation by means of inverse 
squares of wave-lengths, or sqvared wave-numbers. 

This method will be found to give a curve very nearly 
straight, but for the present purpose the increased amount 
of calculation involved in the method renders its use incon- 
venient. 

Method IV. AHthmetical interpolation of wave-lengths. 

It is sometimes possible to find two known lines near to, 
and one on each side of, the line to be measured. Calling the 
wave-lengths of these two lines X, and X^ and their scale readings 
71, and n^, it is seen that a distance of n^ — n^ on the scale corre- 
sponds to a change of wave-lengths X, — X^. Hence if n is the 
reading of the unknown line and X its wave-length, a difference 
n — ni on the scale must correspond to a difference X, — X in 
wave-length, and if change in wave-length is proportional to 
change in reading 

Xi — X n — ni 



Hence 



A«j "^ A^ n^ "^ Tlj 






an equation which we may use to calculate X when the three 
lines are near together, but not in other cases. 

Method V. Arithmetical interpolation by means of wave- 
numbers. 

The use of this method will be understood from what pre- 
cedes. 



192 



LIGHT 



XL 



Method VI. Arithmetical interpolation by means of inverse 
squares of wave-lengths or squared wave-numbers. 

It has already been stated that equal distances on the scale 
correspond very nearly to equal differences in the inverse squares 
of the wave-lengths. Hence calling 3/1, y^ and y the inverse 
squares of the wave-lengths of the two reference lines and of 
the unknown line respectively, and replacing the values of \ by 
the corresponding values of y in the result given in Method IV. 
we have : — 

2' = 2/. + ;^^/2/3-2/.). 

From y, the wave-length is found by calculating Ijs/y. 

When the reference lines are not very close together this is 
the method which should always be adopted when arithmetical 
interpolation is employed. 

Exercise III. Map a spectrum on the normal scale. 

Using the curve obtained in Exercise XL, make as accurate a 
drawing as you can of the calcium spectrum on a scale of wave- 
lengths, representing the characteristic features of the spectrum 
by one of the two methods of mapping spectra described on 
pages 185 and 186. 

Exercise IV. Using the measurements previously made, 
calculate, by Methods IV. and VI. above, the wave-length of 
the sodium line, having given the wave-lengths of the lithium 
and thallium lines. 

Enter observations and results as follows : — 



Line. 


Scale Division. 


Wave-lengths in Xtt metres. 


Given. 


Calculated 
Method IV. 


Calculated 
Method VI. 


Lithium 


6-15 


6708 






Thallium . . . 


10-55 


5351 






Sodium 


8-0 




5932 


5898 



Correct Wave-length = 5893. 



XL 



REDUCTION OF SPECTROSCOPIC MEASUREMENTS 



193 



Notice that Method IV. gives only an approximate value, but 
that Method VI. gives a fairly accurate result. This shews that 
the reference lines are too distant fi*om the unknown line for 
the first method of direct interpolation to be applicable, and for 
very accurate work even the method of inverse squares should 
not be used for lines which are as wide apart as these. 

TABLE. 





Wave-length 


Wave-number 


Inverse Square of 


Reference Lines. 


Xth Metres 


l>er Centimetre 


Wave-length 




in air. 


in air. 


in air. 


K red. Centre of 


7682-5 


13016-6 


1694-3 X 10» 


double line 








Li red 


6708-2 


14907-1 


2222-2 


H. red. (C) 


6563-05 


15236-8 


2321-6 


Na yellow. Cen- 


5893-17 


16968-8 


2879-4 


tre of double 








line (D) 








Tl green. 


5350-65 


18689-3 


3492-8 


Up green (F) 


4861-50 


20569-8 


4231-2 


Sr blue 


4607-51 


21703-6 


4710-5 


Ca blue 


4226-90 


23658-0 


5597-0 


K violet. Centre 


4045-82 


24716-9 


6109-3 


of double line 









The wave-lengths and wave-numbers are here given to six 
significant figures, as the more complete values may be useful 
for reference, but for the purposes of the above exercises it will 
be sufficient to use four figures. The wave-lengths are taken 
as far as possible from Rowland's maps; in the case of Potassium, 
Lithium, and Thallium the numbers are those given by Kayser 
and Runge. 



8. P. 



13 



SECTION XLI. 

THE SPECTROMETER AND ITS ADJUSTMENTS. 



Apparatus required : Spectrometei\ with Gauss eye-piece, 
plane parallel glass on stand. 

An instrument in which the deviation produced by the 
passage of a beam of parallel light through a prism or other 
apparatus can be measured is called a spectrometer. It 
consists of a collimator S, a telescope T, and a horizontal 
divided circle G (fig. 83). The telescope moves about a vertical 
axis passing through the centre of the circle. Attached to 
the telescope is an arm carrying two verniers V, V opposite 




Fig. 83. 

each other, which move along the circular scale, so that the 
angle through which the telescope is turned may be accurately 
measured*. Above the divided scale is a table B, on which 
prisms or gratings may be placed. In some instruments 
* In fig. 83 the circle and verniers are enclosed in a case. 



XLI THE SPECTROMETER AND ITS ADJUSTMENTS 195 

this table is fixed, but ought to be capable of being turned 
about an axis coincident with the axis of rotation of the 
telescope. The way in which an angular displacement of the 
table is measured diffei-s in different instruments. If the 
divided circle is fixed to the collimator, the table should carry 
a second set of verniers. But in many instruments the table 
and divided circle move together. If that Ls the case, care 
must be taken that between any two readings of the verniers 
either the table only, or the telescope only is moved. If both 
are displaced, the vernier readings will only shew the difference 
between the angular displacements of table and telescope, not 
the actual displacement of either. 

It is difficult to secure that the axis of rotation passes 
accurately through the centre of the divided circle. The error 
thereby introduced, called " error of eccentricity," is eliminated 
by having two verniei-s opposite each other as described. The 
angle measured by means of one vernier exceeds the correct 
value by as much as the angle measured on the other falls short 
of it ; so that the arithmetic mean of the two results gives the 
correct angle. For a proof of this see Note, p. 202. 

Examine the scale and vernier, and determine the value of 
the smallest subdivision of the principal and vernier scales. 
If e.g. the circular scale is subdivided into 20 minutes of arc, 
the vernier will probably be divided into 20 parts, of which 
every fifth will be numbered, and each of the 20 may be sub- 
divided into 2 or 3 parts, which will allow the measurements 
to be made to oO or 20 seconds of arc. In entering an obser- 
vation in your Note-book, write down separately the readings 
of the two scales. Thus if the angle to be read off were 
At*!" 43' 20", the observation would be entered as follows: — 

Principal Scale 47° 40' 0" 

Vernier 3' W 

47° 43' 20" 

A w(M^l('u model ot the vernier is placed in the laboratory, 
and the student should practise reading it until he is quite 
familiar with it. 

Before proceeding with the exercises students should study 

13—2 



196 



LIGHT 



XLI 



the construction of the instrument they are using, and refer to 
a more detailed description, which will be supplied with the 
instrument. Special care should be taken to be familiar with 
the object of the various screw heads, which serve either to 
clamp some part of the instrument, or to give that part a slow 
motion. If the telescope, for instance, is to be pointed in any 
direction, it is first moved by hand as nearly as possible into 
its right position. It is then " clamped " by the proper screw, 
and finally brought to the proper position by means of a " fine 
adjustment screw," which can, within certain limits, alter its 
direction. For the purpose of clamping, it is not necessary to 
use any force. If tlie screw is screwed up gently it will he 
sufficient. If force is used the instrument will be damaged. 

Other moveable parts of the instrument will also in general 
be provided with a clamping arrangement and fine adjustment. 

Before moving any part of the instrument care must be taken 
that the clamping screw of that part is released. All parts 
should move easily, and no force shoidd ever be used. 

In order to fix the direction in which the telescope 
points, a mark is placed in the focal plane of the object lens. 
This mark consists generally of a cross formed by two cocoon 
fibres, spider's threads, or very thin platinum wires. This 
arrangement, called the " cross wires," can be turned in its own 
plane, so that one of the wires may be placed vertically, or they 
may both be placed at an equal inclination to the vertical. The 
latter position is most convenient when the telescope is to be 
pointed towards the image of a slit, for it is found easier to. 




Fig. 85. 

place the centre of the cross on the image when the cross has 
the position shewn in Fig. 84, where AB represents the image 



XLI THE SPECTROMETER AND ITS ADJUSTMENTS 197 

of the slit, than when one of the wires is vertical, as in Fig. 85. 
For when the slit is narrow, as it should be, the vertical wire 
will be difficult to see against the dark background. 

When the field of view is dark, it may be necessary to throw 
some light into the telescope in order to see the cross wires 
distinctly. This is best done by means of a sheet of white 
paper held in the hand and placed obliquely near the object- 
glass of the telescope ; the light scattered from the white paper 
will illuminate the field of view sufficiently. 

To focus the eye-piece on the cross wires, there must be 
some means of altering the positions of one of the two. If the 
cross wires are fixed in the draw tube, the eye-piece may be 
moved slightly outwards or inwards, if the eye-piece is fixed the 
wires are mounted on a frame which admits of a small motion. 
Turning the telescope towards a bright surface, such as an 
illuminated sheet of paper, move the eye-piece or the cross 
wires until you see them distinctly. The eye will be least 
fatigued, if the cross wires are so placed that their image 
is as far removed i.e. the eye-piece drawn out as much as 
possible, consistently with distinct vision. 

To focus the telescope turn it towards some distant object, 
and alter the distance between object lens and eye-piece till a 
distinct image of the object is produced at the cross wires. 
Move the head sideways to see that there is no parallax. 

If a parallel beam entei-s the telescope in such a direction 
that it converges to the central point of the cross wires, a line 
drawn through that central point parallel to the original 
direction of the beam, is the " line of sight " of the telescope. 
If the light comes from a star, the line of sight is the line 
drawn from the centre of the cross towards the star. This line 
of sight should be perpendicular to the axis of rotation of the 
telescope (see p. 199). 

In order to adjust the collimator to give a parallel beam of 
light, tuni the telescope so as to look straight into the collimator 
tulie. Place a luminous burner 10 cms. behind the slit of thr 
latter and focus the telescope until the edges of the slit look 
perfectly sharp. Place the cross wires on it, move the eye to 
the right or left and see that thfere is no parallax. 



198 LIGHT XLI 

The level of the collimator must now be fixed, so that a 
point near the centre of the slit shall have its image covered by 
the cross wire. For this purpose a fine wire or thread should 
be stretched across the centre of the slit. If the level of the 
collimator admits of alteration, it should be adjusted until the 
image of this thread coincides with the centre of the cross. If 
the collimator is fixed, the position of the thread on the slit 
must be altered until the condition is nearly satisfied, great 
accuracy of this adjustment not being important. 

Another method of focussing telescope and collimator is 
sometimes found more convenient, 
as it does not involve removing the v 

prism or other apparatus from the ^', ^ j^ 

table. This method is based on the i ,.^--<^i^^^^^^~"^<> 
fact, that if a beam of light diverging S |-^*''^^^^^ 
from a point S falls on a prism, the ' 
distance of the virtual image S' ^^^- ^^• 

from the prism is greater the greater 

the angle of incidence of the original beam. If the prism is 
placed in the position of minimum deviation (Fig. 86), the 
virtual image will be at the same distance as the object, but as 
the prism is turned in the direction of the hands of a watch, 
the virtual image will move further away. If the distance of 
S from the prism is very great, so that a parallel beam of light 
falls on the prism, the rays will remain parallel after refraction, 
whatever the angle of incidence. If the original beam is 
parallel therefore, and only in that case, will the position of 
the prism have no effect on the distance of the virtual image from 
the observer. This furnishes a delicate test of the parallelism of 
a beam of light. If the light is not parallel the adjustment 
may be made as follows : — 

Place a prism on the spectrometer table approximately in 
the position of minimum deviation. Illuminate the slit with 
sodium light and turn the telescope so as to see the sodium 
line. Fix the position of the telescope so that when the prism 
is turned round, the image moves across the field of view and 
comes to minimum deviation just at its edge or a little beyond. 
It is clear that there will now be two positions of the 



XU THE SPECTROMETER AND ITS ADJUSTMENTS 199 

prism for which the sodium line will be in the centre of the 
field of view. In fihe first position, which we may call the 
" slanting position," the incidence of the beam on the prism is 
greater than at minimum deviation, while in the second which 
we may call the " normal position " the incidence is smaller. 
Rotate the prism to the slanting position, and focus the 
telescope carefully until you see the edges of the slit quite 
sharply ; it is better for this purpose not to have too narrow a 
slit. Now turn the prism into the normal position. If the 
focus is still good, the collimator supplies a parallel beam. 
But if the image is out of focus, this shews that the collimator 
is not in adjustment, and as we know that the change of 
position of the prism must have brought the image nearer, we 
may remove it to its old position by adjusting the collimator. 
When this is done the prism is turned bacj^ to the slanting 
position. In doing so we remove the image still further, and 
must focus again, but this time with the telescope. 

Repeat the process several times, taking care always to focus 
the telescope tuhen the prism is in the slanting position, and the 
collimator when the prism is in the normal position. After 
three or four changes it will generally be found that the image 
remains in focus during the displacement of the prism, and the 
adjustment is then complete. 

If the telescope is now directed straight towards the 
collimator, the image of the slit should be in focus at the 
cross wires, and there should be no parallax on moving the eye 
to the right or left. When the prism is in use, if the faces of 
the prism are not quite plane, it may happen that the focus is 
slightly different according as light passes through the prism, 
or is reflected from one of its faces. Unless the image is 
rendered markedly indistinct it is best, however, not to alter 
the focus. The prisms generally supplied with spectrometers 
of ordinary size are sufficiently plane not to require any 
readjustment of focus. 

We have seen that the line of sight of the telescope should 
be at right angles to the axis of rotation, i.e., parallel to the 
plane of the divided circle. 

This adjustment is more difficult to carry out. Fortunately 



200 



LIGHT 



XLI 



the errors introduced, when the condition is not accurately 
fulfilled are not very great, and unless greaBaccuracy is required 
it need not be carried out. The m^thod^ instructive however, 
and should be understood. ^ 

In Fig. 87 let AB he the axis of rotation, LM the line of 
sight of the telescope. Also let UK be a piece of plane parallel 
glass; placed so that a ray of light passing along ML is reflected 
back along the same line LM, i.e. with HK at right angles to 




Fig. 87. 

LM. If now the arm OGD is rotated through two right angles 
round AB, until the line of sight comes into the position L'M\ 
an inspection of the figure will shew that the surface HK will 
no longer be at right angles to L'M\ unless the line LM was 
originally at right angles to AB. 

It remains to be shewn how the line of sight of the 
telescope may be adjusted to satisfy this condition. The 
telescope is provided with an eye-piece having an aperture 
at the side, through which light may fall on an inclined piece 
of glass inserted between the two lenses of the eye-piece. The 
light from a luminous burner is reflected from the glass plate, 
and illuminates the cross wires (Fig. 88). 




^^ 



Fig. 88. 



The light diverging from these wires is converted by the 
object-glass of the telescope into a parallel beam of light, 



XLI THE SPECTROMETER AND ITS ADJUSTMENTS 201 

which is reflected from the plane surface HK along its own 
path, if that surface is at right angles to the line of sight of 
the telescope. An image of the cross wire will be formed, 
which will coincide with the wires themselves if all the 
adjustments are perfect. A slight displacement of HK will 
displace the image of the cross wires, and unless HK is 
approximately in the right position, the image may not appear 
in the field of view at all. In making the adjustment proceed 
as follows. Place the glass plate on the table of the spectro- 
meter so that the plane of the plate is perpendicular to the 
line joining two levelling screws of the table, and loosen the 
clamping screws of table and telescope. Turn the table until 
the line of sight of the telescope is, as far as you can judge by 
the eye, at right angles to the glass plate. Place your eye near 
the eye-piece of the telescope, and look at the image of the 
telescope in the glass plate. This image should be in line with 
the telescope itself as in Fig. 89. 





Fig. 89. Fig. 90. 

If the image points downwards, as in Fig. 90, the plate 
should be tilted backwai-ds by means of the levelling screws. 
A little practice will enable you to set the plate nearly right 
in this way by eye. When this is done, clamp the table, place 
a light at the side of the eye-piece, and look through the 
telescope. A small angular movement of the telescope to the 
right or left; should now bring the image of the cross wires into 
sight. Now level the glass plate till on moving the telescope 
the image of the cross wires coincides exactly with the wires 
them.selves. 

The adjustment having been made on one side, turn the 
telescope* through two right angles, and see whether the image 
of the cross wires formed by reflection from the other side of 
the plate coincides with the wires. If this is the case, the line 



202 



LIGHT 



XLI 



of sight is at right angles to the axis of rotation as required. 
If not, bring the two images into coincidence by first tilting the 
telescope until their distance apart has been halved, and then 
tilting the glass plate until the images are in coincidence. 
Turn the telescope again through two right angles, and if 
necessary repeat the adjustment till the images coincide with 
the cross wires in both positions of the telescope. 

If the two images of the cross are not both sharply defined, 
the telescope has not been properly adjusted for parallel rays, 
or the glass plate is not perfectly plane. 



In 



Note. 

In Fig. 91 let G be the centre of the divided circle, 
any position of the telescope let one of 
the verniers be at A and the other at A\ 
and let be the axis of rotation of the 
telescope. Let the telescope be rotated 
till the verniers are at BB\ Then BOA 
will be the angle through which the 
telescope has been rotated, while BCA 
and B'CA' will be the apparent angular 
displacements of the two verniers as read 
off on the divided circle. 
But 

BCA = 20B'A = BOA + OF A - OAF 
and 

B'CA' = 20 AF = BOA + OAF - OF A 
Hence by addition BCA -{-B'CA' = 2B0A 

which shews that the ^ngle thi'ough which the telescope is 
rotated, is correctly obtained by taking the arithmetic mean of 
the angles read off on the vernier scale. 




SECTION XLII. 

DETERMINATION OF THE REFRACTIVE INDEX OF 
A SOIJD BY THE SPECTROMETER. 

Apparatus required : Spectrometer in adjtcstmerUy glass 
prism, small mirror and reading lens, sodium bead and flame. 

The method used in determining the refiuctive index of a 
solid is identical with that explained in the elementary exercise * 
on the same subject, with which the student is supposed to be 
familiar. The first step is to determine the refracting angle of 
the prism. For this purpose the prism must be placed on the 
table of the spectrometer so that its two refracting fiices, and 
therefore their intersection, are perpendicular to the plane of 
the graduated circle. The table of 
the spectrometer ought to consist 
of a platform which can be levelled 
independently of the circular scale. 
If this is not the case, the prism 
should be placed on a separate 
support provided with levelling 
8cre^vs. 

In Fig. 92 PQR are the three *'»«• ^^' 

levelling screws, and the prism is placed in the centre of the 
table in such a way that one of the faces, say AC, is at right 
angles to the line joining two screws, say R and Q. This 
may be done with sufficient accui-acy by eye. To adjust the 
faces of the prism, place the collimator and telescope at a small 
angle to each other, illuminate the slit and turn the table of 
the spectrometer until an image of the slit, formed by reflection 
at AC, appears in the centre of the field. 

• IfUermediaU Practical Phy$icM, Sec. xxviii. 




204 LIGHT XLII 

The slit should be sufficiently narrow to allow you to place 
the cross accurately over its image ; but it is not necessary to 
have it as nan-ow as you would use it in spectroscopic work, 
unless you desire to resolve the double sodium line and measure 
the refractive index for each line separately. 

The spectrometer table must now be levelled so that the 
image of a thread stretched across the centre of the slit 
coincides with the centre of the cross wires of the telescope. 
The table is then turned till the reflection of the slit is 
obtained from the face AB, and the image of the thread made 
to coincide with the centre of the cross wires by levelling the 
prism by means of the screw P, which is the only one that does 
not alter the inclination of the face AG, but only turns that 
face in its own plane. After the face AB has been levelled in 
this way, return to the face AG; if you have not originally 
succeeded in placing AG at right angles to QR, you may have 
slightly altered the inclination of the face. If so, set it right 
again by the screw Q which least disturbs the face AB. 

Going backwards and forwards once or twice will always 
enable you to secure that the two faces are both properly 
adjusted. 

Proceed next to measure the angle of the prism. Keep 
the telescope fixed, and turn the prism so that the intersection 
of the cross wires lies exactly on the image of the slit formed 
by reflection at one face of the prism. Read both verniers. 
Next turn the prism till the image of the slit formed by the 
second face is on the cross wire, and again read both verniers. 
If the table has been turned through an angle 0° between the 
two observations, the angle of the prism is 180° — 6°. 

If in moving from one position to the other, a vernier 
passes over the zero of the scale, 360° must be added to the 
smaller reading, and the higher reading subtracted from it. 

Alter the position of the telescope by a few degrees, and 
repeat your observations, obtaining a second value for the angle 
of the prism, which should agree closely with the first. 

This method implies that the table of the spectrometer is 
moveable. If it is not so, the method described in the Note 
p. 207 must be adopted. 



Vernier A. 




Vernier B. 


350° 6' 0" 




170° 2' 20" 


110° 13' 0" 




290° ir 0" 


120° r 0" 


120° 8' 40" 


... *- 120° 


r 


50" 


59° 


52' 


10" 



XLII REFRACTIVE INDEX OF A SOLID 205 

Enter your observations as follows : 
Jan. 24, 1894. 
Spectrometer J., Prism A. 
Determination of Angle of PHsm. 

First position of prism 
Second „ 

Difference ... 

Mean 

.*. Angle of prism ... 

Similarly for the second measurement. 

Having determined the angle of the prism proceed to find 
the minimum deviation for light of the kind for which the 
refractive index of the prism is required, e.g., sodium light. 
Place a Bunsen burner with a sodium bead in it behind the 
slit, and without altering the adjustment of the prism turn the 
spectrometer table so that the light coming from the collimator 
is refracted through the prism. Find the refracted image of 
the slit in the first place by eye, and follow the image while 
the prism is slowly turned round till the position of minimum 
deviation is found roughly. When this is done bring the 
telescope into this position, and find the image of the slit in 
the telescope. Watch this image while the table of the spec- 
trometer is slowly turned. If the direction of rotation is 
))roperly chosen, you will find the image moves slowly in 
the direction of deviation, comes to a standstill, and moves 
back again. Leave the prism in the position for which 
the deviation is least, place the cross wires of the telescope 
approximately on the image, clamp* the telescope and adjust 
the cross wires more accurately by means of the slow motion. 
Now tuni the prism again backwards and forwards, making 
sure that the centre of the image of the sht just comes up to 
the centre of the cross wires, but does not pass beyond it. 
Adjust the telescope if nece.ssary until you are quite satisfied 
that this is the msv. Rriul both Verniers. 

' Sro ifiuuiks lis t(j clamiung, Ac, p. i'.M). 



206 LIGHT XLII 

Now clamp the table of the spectrometei'. Remove the prism, 
turn the telescope so as to point directly towards the colli- 
mator, and adjust it until the image of the slit is bisected by 
the cross wires. Read the verniers to determine the position 
of the telescope when there is no deviation of the ray. The 
difference in the readings in this position and in the position of 
minimum deviation gives the angle of minimum deviation. 

It is advisable to obtain two independent readings of the 
position of minimum deviation so as to secure accurate results. 

When the spectrometer table is fixed, it is not material 
whether the direct reading is taken before that of minimum 
deviation or vice versa. But when the graduated circle moves 
with the spectrometer table the above order mitst be adhered to. 
For it has already been pointed out that correct values will in 
that case only be obtained if either the telescope only, or 'the 
table only, is turned round between two readings. As the 
adjustment to minimum deviation necessarily involves the 
turning round of the table, this should be done first and the 
table be clamped before the telescope is moved round to take 
the direct reading. 

The observations are entered as follows : — 

Detertnination of Minimum, Deviation. 





Vernier A. Vernier B. 


Prism at minimum deviation 


143° 40' 00'' 323' 40' 40" 


Direct reading 


94° 13' 00" 274° 14' 20" 




49° 27' 00" 49° 26' 20" 


Angle of minimum deviation 


49° 26' 40" 


Similarly for the second experiment. 


Angle of minimum deviation 


49° 27' 40" 


MeanD 


49° 27' 10" 


Angle of prism a 


59° 52' 10" 


D + a 


= 109° 19' 20" 


(D + oi)/2 


= 54° 39' 40" 


a/2 


= 29° 56' 05" 


. n + a 

sm — ^^ 




^.aoA^ 


a 
sm 2 



XLII REFRACTIVE INDEX OF A SOLID 207 

The observations should be suflSciently accurate to give the 
third decimal place with certainty, and the fourth decimal 
place with fair accuracy. The numerical calculation should be 
carried out with a table of seven figure logarithms. 

Calculate the error produced in n on the supposition that 
an error of half a minute has been made, (a) in the measure- 
ment of the angle, (6) in the determination of the minimum 
deviation. 

Note. 

If the table of the spectrometer is not moveable, the angle 
of the prism may be measured as follows: — Place the prism 
with its refracting edge towards the collimator in such a 
position, that about half the beam of 
light falls on one and half on the other 
face of the prism. A reflected image 
may now be obtained from each face, and 
if the telescope is first placed so as to 
point to one image, and then turned until 
it points to the other, the angle through „. ' ' 

which the telescope has been turned is 
twice the angle of the prism. In Fig. 93, let / be the virtual 
image of the slit formed by the collimator, ABC the prism, /, 
and Li the two images formed by reflection at the faces AB 
and AC oi the prism. By the laws of reflection we have the 
angles 

lAI, = 2IAN, 

lAI, = 2IAN, 

.'.I,AI,--2N,AN, = 2a. 

The images /, and /j therefore subtend an angle 2a at the 
refracting edge A, and if the telescope is pointed first towards /, 
and then towards I^, the angle through which it turns will be 
2a, if the axis of rotation of the telescope is at A, or if / is at 
an infinite distance. Hence, in order that this method should 
give correct results, the collimator ought to be well adjusted 
for parallel light, and to correct any error due to a faulty 
adjustment, the refracting edge should be placed nearly over 




208 LIGHT XLII 

the axis of rotation of the telescope. The errors introduced may 
otherwise be quite appreciable. If, for instance, the distance 
between the refracting edge and the axis of rotation is one 
centimetre and the image of the slit at a distance of 50 metres, 
which is quite possible if the collimator is of ordinary size, the 
error may amount to over a minute of arc. 

Quite apart from this source of error, which might be 
avoided, the method is not a good one, for each image is 
formed by an unsymmetrical beam of light, filling at the 
utmost only half the telescope lens, and serious errors may 
creep in due to the aberrations of the lenses. 



SECTION XLIIL 

DETERMINATION OF THE INDEX OF REFRACTION 
OF A LIQUID. SPECIFIC REFRACTIVE POWERS. 

Apparatus required : Spectrometer, hollow glass prisin, 
alcohol, theinnometer, Bunsen flame, and sodium head. 

When the index of refraction of a liquid is to be determined, 
the liquid is placed in a hollow prism the vertical sides of 
which are two plates of glass. If these glass plates are accu- 
rately plane parallel they produce no deviation of the rays of 
light passing through them, and the deviation observed is due 
to the liquid only. Hence the angle of the prism of liquid may 
be measured, and the refractive index determined by the devia- 
tion, just as in the case of a solid glass prism. If the plates 
of glass are slightly prismatic their effects on the apparent 
angle and deviation must be eliminated. To enable this to be 
done they are not cemented to the rest of the prism, but are 
made moveable, and are kept in position by rubber bands or by 
clips. After observations of the angle of the prism and of the 
deviation have been made each plate is rotated about its normal 
through 180^ and the observations repeated. The means of the 
two sets of observations of the angle and of the deviation are 
the required angle and deviation of the liquid prism. 

Clean the prism provided, and fill it with absolute alcohol 
at the temperature of the room. 

As the prism may leak a little, it is not placed directly on 
the table of the spectrometer, but in a shallow metal dish which 
stands on three short legs on the table, and itself supports the 
prism at three points. 

The adjustment of the prism and the determination of the 
angle and deviation are made as in the previous section. In 

8. P. U 



210 LIGHT XLIII 

determining the angle the stronger of the two reflected images 
of the slit seen in the telescope should be used in each case. 
It is formed by reflection at the air-glass surface, while the 
weaker is formed by reflection ai the glass-liquid surface. 

The glass plates forming the sides of the prism are then 
rotated as described above and the observations repeated. Each 
plate is provided with a mark, the position of which should be 
recorded, along with the observations. 

The mean deviation D and the mean angle a are then 
found and the refractive index calculated from them in the 
usual way. 

Enter your observations and results as follows : — 

15 October, 1895. 
Prism 1 1. Absolute Alcohol. 
Determination of deviation produced by prism when the 
marks on the glass plates are near angle of prism : — 

With prism 
Without „ 

Apparent deviation 
Determination of angle of prism when the marks on the 
plates are near angle of prism : — 

First position ... 246° 29' 30" 66° 29' 30" 

Second „ ... 126 29 306 29 30 

Difference ... 120 30 120 

Mean 120° 0' 15". 

.'. Apparent angle of prism = 59° 59' 45". 

Give similarly the readings of the verniers when the plates 
have been rotated and the marks are near the base of the prism, 
and collect the results as follows : — 

Mean minimum deviation = D 

Mean angle of prism = a 

. • . Refractive index 

Temperature of alcohol 

Refractive index calculated from re- 



Vernier A. 




Vernier B. 


229° 5' 0" 




49° 4' 30' 


23 9 




203 7 30 


25° 56' 0" 




25° 57' 0" 


... 25° 


56 


;' 30" 



25° 


36' 30" 


59 


51 42 




1-3635 


... 


18° C. 



f ,. , 1-365 

fractive powers 



XLIII REFRACTIVE POWERS 211 

If greater accuracy is requii-ed several measurements should 
be made and the mean of the results taken. 

The observed refractive index may be compared with the 
index calculated by a method founded on the following facts : — 

If fi is the refractive index and d the density of a substance, 

the value of the expression , is called by Gladstone the 

specific refractive power of the substance. It is not sensibly 
altered by variations of temperature or pressure, but changes 
when the substance passes from the liquid to the gaseous state. 
Lorenz and Lorentz have from theory deduced another expression, 

namely, -, — ^--t. which remains nearly constant, even when 
/I* 4- 2 a 

the substance changes its state. For the purpose of this 
exercise it will be sufficient to use the former expression, as 
it more easily lends itself to numerical calculation. The refrac- 
tive power varies with the wave-length of the light used, and 
we shall take ^ to refer to soilium light. 

It is found by experiment, that the refractive power of a 
mixture of substances is equal to the mean refractive power of 
the constituents. Thus, if p^, p^, p^, &c. grams, of a number of 
liquids are mixed together, we have 

where P = pi-\-p.,-\-p:i+ ... , and (N — I )/D is the refractive 
power of the mixture. 

It has also been found, that for a compound, the molecule M 
of which contains n^ atoms of an element the atomic weight of 
which is m,, /i, atoms of an element the atomic weight of which 
is jti-i, and so on, the weights of the different elements in the 
combination being therefore Tiiniit w,?/*,, etc., we have 

iX ^ \ uu — 1 ii. — 1 

If we call ^ J - the atomic refractive power of the first 

element, and designate it by r^, and so on for the other elements, 

14—2 



212 



LIGHT 



XLIII 



f,-l 



a 



the molecular refmctive power R of the compound, 
we have then the equation 

R = 7ii?'i + iur._, + &c., 



or in words : — the refractive power of a compound is the sum of 
the products formed by multiplying the atomic refractive power 
of each element by the number of atoms of that element con- 
tained in the molecule of the compound. 

We cannot dii*ectly determine the atomic refractive power 
of an element entering into a compound, but we may find it in- 
directly. If, for instance, we take three compounds containing 
carbon, hydrogen, and oxygen in different proportions, we obtain 
three equations to determine ?\, r^, and r^ the atomic refractions 
of carbon, hydrogen, and oxygen respectively, and if we find that 
the refractive indices of other compounds may be determined 
with the help of the values so found, we shall have justified the 
above statement. It appears, however, that the way in which 
an atom is combined affects its refractive power, and to obtain 
consistent results it is necessary to assume, for instance, that 
oxygen in the carbonyl group has a different refractive power 
from the hydroxyl oxygen. 

Calculate the refractive index of ethyl alcohol, making use 
of the following numbers : — 

Atomic refraction of ^ ... ... ... 1*66 



„ „ of in alcohols 

ofC 
Density of ethyl alcohol, at 20° = 
Coefficient of cubical expansion = 

Calculated index of alcohol at 18° C. = 
Measured „ „ = 



2-68 

4-36 
•788 
•0010 

1-365 
l-3(>3 



SECTION XLIV. 

PHOTOMETRY. 

Apparatus required : Photometer, scale, candles, and gas- 
lamp. 

The photometer is an instrument for comparing the illu- 
minating powers of different sources of light. One of the 
most accurate forms of the instrument is that in which the 
two sources are placed on opposite sides of a screen capable of 
partly reflecting and partly transmitting the light from each 
source. When this screen is placed in such a position that the 
straight lines drawn from it to the two sources make equal 
angles with the screen, and the illumination of the two sides of 
the screen appears the same, the amounts of light to be 
compared are to each other as the squares of the distances of 
the sources from the screen. 

Having fixed on the light sent out by some standai*d source 
in a given direction as the unit, we can by a comparison of any 
source with the standard determine the illuminating power of 
that source. 

The standard source is a .sperm candle flame consuming 
120 grains of sperm per hour, and the standard direction any 
line in a horizontal plane. If the consumption of sperm differs 
from the stjindard mte by less than 10 grains per hour, the 
illuminating power may be taken as proportional to the con- 
sumption. 

Arrange the two sperm candles, the photometer, and the 
gas flame provided, in a straight line in the order named, the 



214 LIGHT XLIV 

two candles being mounted about 5 cms. apart in a direction 
perpendicular to the straight line, on a stand capable of moving 
along a graduated scale parallel to the line, Fig. 94. Place the 





Fig. 94. 

gas flame at some convenient fixed distance from the photo- 
meter screen, say 1 metre, with a flat surface towards the 
screen, light the candles and after a few minutes move them 
backwards and forwards along the scale, till the illumination 
of the two sides of the photometer screen is equal. If the 
simple Bunsen screen is used, this is the case when the semi- 
opaque and semi-transparent parts of the screen appear of equal 
brightness, and if the Lummer-Brodhun instrument^ is used, 
when the inner and outer parts of the field of view of the 
telescope through which the screen is viewed appear of equal 
brightness. If it is not possible to produce equality of bright- 
ness without moving the candles inconveniently near to or far 
from the photometer, increase or decrease the distance of the 
gas flame. 

Blow out the candles, weigh and replace them, and at a 
given instant relight them. Move them till equality of 
brightness of the screen is produced, and determine the 
distance of the candles from the screen by reading the scale 
position of a point vertically under the candles. If the 
photometer screen can be reversed this should be done and 
equality of illumination again secured by moving the candles. 
Their position should again be read on the scale. These 
observations should be repeated three or four times and the 
means taken. 

Rotate the gas flame so that the other flat surface is now 
presented to the photometer, and repeat the observations, 
taking the means. 

^ This is the instrument shown in Fig. 94. 



XLIV 



PHOTOMETRY 



215 



In the same way take observations when the edges of the 
flame are presented to the photometer. 

Blow out the candles, noting the time, and again weigh 
them. 

Record as follows: — 

19 Jan., 1897. 

Tulip gas burner. 

Distance D of gas flame from photometer screen = 80 cms. 
Weight of candles when lighted at lOh. 40m. 154*2 grams. 
„ blownoutatllh.lOm. 1460 .. 



Sperm consumed in '5 hrs. 



( 8-2 gms. 



127 



127 grains. 



.*. Candle power of candles = X= ^ ' ^ = 2*12 standard 
^ 5 X 120 



candles. 



Gas flame 


Mean 

distances of 

candles from 

screen=<i 


(? 


D» 


D*ld* 


-? 


Candle 
power 


Flat (1) { 


40cma 
38 „ 


1600 
1524 


6400 


40 
4-2 


8-48 
8-91 


18-69 


.. (2) { 










' 


■ 


Edge(l) { 






Ac. 






} 


.. (2) j 












I 



SECTION XLV. 

INTERFERENCE OF LIGHT. 

Apparatus required : Bunsen flame, sodium bead, slit, 
biprism, micrometer microscope, metre scale. 

When two beams of homogeneous light coming from the 
same source cross each other after having described paths 
differing in length, the vibrations due to the two may be in 
opposite directions and neutralise each other at certain points 
of the region where the beams cross. At such points the joint 
action of the two beams will produce partial or total darkness, 
and if a screen is placed in this region, a series of light and dark 
" interference " bands will be seen on it. 

In the case in which the source of light is a narrow vertical 
slit behind which a sodium flame is placed, and the two beams 
are produced by the passage of the light from this slit 
through a double prism the section of which is indicated 
in Fig. 95, the distance x between consecutive dark 
bands on the screen, is related to the wave-length X of 
the light, the distance a of the slit from the screen, and 
the distance apart c of the two virtual images of the slit Fig. 95. 

formed by the biprism, by the equation : X = -— . 

This method of producing interference bands may therefore 
be used to determine approximately the wave-length of light. 

The experiment is performed on an " optical bench," a 
simple form of which is shewn in Fig. 96. 

The slit consists of two brass plates with straight bevelled 
edges, attached by screws to a wooden stand provided with 
levelling screws. 



XLV 



INTERFERENCE OF LIGHT 



217 



The brass plates should be adjusted so that their edges are 
about 1 mm. apart and parallel to each other. This is most 




Fig. 96. 

easily done by placing a strip of thin writing paper between 
the edges, pressing the plates together and screwing them 
firmly to the wood, and then removing the paper. 

The biprism is mounted on a similar stand. 

The micrometer microscope consists of an eyepiece mag- 
nifying a few times, across the focal plane of which a cross wire, 
or by preference a pointed vertical needle, can be moved by 
means of a screw with a graduated head on which the number 
of turns can be read. The arrangement is mounted on a heavy 
wooden base. 

The three stands described above slide along a wooden 
bench provided with a groove in which two levelling screws 
of each stand slide, and with a gniduated scale. 

Place the slit vertical at one end of the stand, and the sodium 
Hame behind it, about 15 cms. in front place the biprism, 
and about 45 cms. in front of the biprism the micrometer eye- 
piece. If the slit, edge of biprism, and centre of field of view 
of the eyepiece, are in a line, and the edge of the prism is neixrly 
pamllel to the slit, a series of light and dark vertical bands 
should be seen on looking through the eyepiece. If no bands 
are visible move the biprism backwards and forwards along the 
bench and tilt it slightly in its own plane till bands are 
visible. Now watch the bands and continue the tilting by 
means of the single screw at one side of the biprism stand. 
Stop the tilting when the bands are most distinct. The edge 



218 



LIGHT 



XLV 



of the biprism will now be parallel to the slit. The biprism 
and eyepiece should now be moved along the bench till the 
bands are both distinct and at a convenient distance apart for 
measurement. When this is the case adjust the reference 
mark in the focal plane of the eyepiece to the centre of a light 
band, read the screw head, move to the next band, read, and 
repeat the process for 5 bands to the left of the centre, then 
omit 5 bands, and read the positions of the next 5. 

Place a metre scale over the slit, biprism, and eyepiece 
stands, and determine their distances apart. 

Record as follows : — 

2 May, 1900. 
Sodium Flame. 



Band 


Beading 


Band 


Beading 


1 
2 
3 
4 
5 


37-72 
38-12 
38-54 
38-84 
39-25 


11 
12 
13 
14 
15 


41-48 
41-85 
42-20 
42-59 
42-93 


Mean 


38-494 




42-210 


Difference 

1 




3-716 





Difference for 10 bands 

1 scale division 

.-. Distance apart of bands 



= 3-716 scale divisions. 
= '05 cms. 
= •0186 cms. 



To determine the distance apart of the two virtual images of 
the slit formed by the biprism, place a lens of about 7 cms. 
focal length, provided with a stop of about '5 cm. diameter, in 
such a position between the biprism and eyepiece, that the 
two images of the slit are in focus. Measure the distance of 
the images apart by the micrometer, and the distances of the 
lens from slit and eyepiece by the metre scale. If a second 
position of the lens can be found for which the images are 
again in focus repeat the observations. 



XLV 



INTERFERENCE OF LIGHT 



219 



The virtual images, when formed b}' prisms of small re- 
fracting angles for nearly normal incidence, are at the same 
distance from the prism as the object. If therefore u is the 
distance of the slit from the lens, it will also represent the 
distance of the virtual images from the lens. The measured 
distance between the real images in the focal plane of the eye- 
piece being c,, the distance between the virtual images will be 

c u 
given by c = -^ , when v is the distance between the lens and 

the focal plane of the eyepiece. For a second position of the 
lens, considerations of elementary optics shew that u and v 
will be interchanged. 

Either of these observations will give c, but the measurement 
of u and v may be avoided altogether, for if we multiply the 
equations together we have c = \UiC^. 

Arrange observations and work as follows : — 



Distance of lens from 


Distance of images 
apart 




8Ut = tt 


screen =r 


Scale 
divisions 


cms. 


c 


19-7 cm. 
40-2 


403 cm. 

19-8 


780 
193 


•390 
•0915 


•191 
•189 
•189 








Mean 


•190 



u-\-v = 60*0 cms. = a. 

xc 0186 X 190 



•'•^''-a" 60 



•000,0580 = 5-80 x lO"" cms. 
Known wave-length of sodium light = 5'893 x 10~' cms. 



SECTION XLVI. 

MEASUREMENT OF THE WAVE-LENGTH OF LIGHT 
BY THE DIFFRACTION GRATING. 

Apparatus required : Spectrometer, diffraction grating, 
Butisen flame, and sodium head. 

When a beam of light diverging from a point passes 
through a transparent plate on one surface of which a series of 
equidistant fine parallel lines has been ruled, the emergent light 
appears to diverge from a number of virtual images, one of which 
is coincident with the luminous point, the others, which lie on 
either side of the luminous point, are called diffraction images. 
The surface on which the lines are ruled is called a diffraction 
surface or grating. 

The connection between the relative positions of the vat-ious 
images, the distance apart of the lines, and the wave-length of 
the light transmitted, can be calculated, and experimental 
observation will therefore furnish a means of measuring the 
wave-length of light. 

In the case of a thin parallel plate of a transparent medium 
with equidistant parallel lines ruled on one surface, if the light 
incident on the ruled surface is a parallel beam, each diffracted 
beam will also be a parallel one, and will be deviated by the 
angle 6, where n\ = b sin 6, n being an integer called the order 
of the image, and h the distance of the lines apart measured from 
centre to centre. Hence if the grating is placed on the table 
of a spectrometer, the collimator of which furnishes a beam of 
parallel light, and the telescope of which is focussed for parallel 
rays, the whole of the images will in turn be visible through 



XL VI THE DIFFRACTION GRATING 221 

the telescope as it is moved from one side to the other. The 
deviation of each diffracted beam may therefore be deter- 
minated on the graduated circular scale of the instrument. If 
a narrow illuminated slit is used instead of a luminous point, 
the images will be linear. 

Adjust the spectrometer as described in section XLI. 
Place a sodium flame behind the slit. Using the Gauss 
eyepiece in the telescope direct it towards the collimator, 
adjust the centre of the cross wires to the image of the slit,, 
and clamp the telescope. 

Place the diffraction grating on the table of the instrument 
with the plane of the ruling perpendicular to the line joining 
two of the screws of the table. 

Rotate the table till the image of the cross wires formed by 
reflection at the surface of the gmting is seen in the field of 
view. Adjust the table by means of the levelling screws till 
the reflected and direct images of the wires coincide. The 
gloating is then perj^endicular to the axis of the collimator and 
telescoj>e. 

Remove the Gauss and substitute an ordinary eyepiece. 
Rotate the telescope till the first diffracted image is on the 
cross wires. If it is indistinct the screw of the table which 
will tilt the lines of the grating in their own plane, should be 
rotated till the image is as distinct as possible. The width of 
the slit should then be varied till the two sodium lines are 
resolved quite distinctly. 

Adjust the telescope by means of the tangent screws till 
the centre of the cross wires coincides with one of the lines, 
and read the vernier of the circular scale. Adjust to the other 
line and again read. Now rotate the telescope, adjust to the 
direct image and read the circular scale. Then rotate further 
in the same direction till the first diffi-acted image on the other 
side is .seen, adjust and reiid the scale for each line. Again 
return to the direct image and take rciulings. By rotating 
still further in each direction, the 2nd, 3rd, &c. images may be 
found and their positions read. 

To obtain the wave-length of light the distance b between 
the lines must be known. In an absolute determination the 



222 



LIGHT 



XLVI 



most difficult part of the measurement consists in finding this 
distance. It may be determined by means of a microscope, but 
in the present exercise b will be assumed to be known. 
Record the observations as follows : — 

6 Jan., 1899. 
Spectrometer A. Grating A. 
Most refrangible Sodium line. 



1 


Vernier A 


Vernier B 


First order, right 


202° 11' 30" 


22° ir 00" 


Direct reading 


179 10 30 


359 10 00 


First order, left 


156 9 30 


336 9 30 


Deviation, right 


23 1 00 


23 1 00 


left 


23 1 00 


23 30 


„ mean 


23 1 00 


23 45 



Mean deviation for first order spectra ... 23° 0' 52". 

Similarly for the other line, and for the spectra of the 2nd 
and higher orders. 

Tabulate results as follows : — 



Grating space of A 



6 Jan., 1899. 

= 6 =00015062 cms. 



n 


Angle of deviation 
= d 


sin e 


sin ^ 1 sin 6 
n n 


1 
2 


23° 0'52" 
51 25 


•3909 
•7817 


•3909 
•3908 


•000,05888 
•000,05887 








Mean 


•000,05888 



If the telescope is now turned to one of the first order 
images, and the table on which the grating is placed is slowly 
rotated, it will be seen that the position of the image, and 
therefore the deviation, change. A position of the grating can 
be found for which the deviation is a minimum. Adjust the 
telescope on the image when least deviated, and read the 



XLVI 



THE DIFFRACTION GRATING 



223 



veraiers. Keeping the table fixed, rotate the telescope till 
the direct image is seen. Read the verniers. Now rotate 
table and telescope till the minimum position of the first 
image on the other side is found, and repeat the observation 
of the deviated and of the direct image. 

The minimum deviation </> is connected with X and b by the 

equation 2b sin ^ = nX, hence X may again be determined. 

Tabulate as follows : — 



First order spectra. 

Vernier A. 

202° 42' 
1 80 8 30 

= 22° 31' 30 



Vernier B. 

22^^ 42' 30' 

10 



22° 32' 30' 
22° 32' 0" 



Deviated reading right 
Direct „ 

Deviation to right 

Mean 

Deviated reading left ... = 
Direct „ ... = &c. 

Deviation to left ... = 

Mean ... ... = 

Mean deviation for first order spectra = 22° 32' 15". 

Similarly for the second order spectra, collecting the results 
in tabular form as follows : — 

6 January, 1899. 



n 






2 


.»! 


^-1 


f-l- 


1 
2 


22- 32' 15" 
46 2 


ir 16' 7" 

23 1 


•1954 
•3910 


•1954 
•1955 


•000,05887 
5889 










Mean 


•000,6888 



By using the grating so that the diffraction images are 
formed by reflected rays, further determinations of X may be 
made. 



SECTION XLVII. 

ROTATION OF PLANE OF POLARISATION. I. 

Apparatus required : Two NicoVs prisms, two tubes with 
glass ends, a solution of sugar, and a sodium flame. 

When a beam of plane polarised light is transmitted 
through certain solids, liquids, and solutions, it is found that 
the plane of polarisation is rotated through an angle pro- 
portional to the length of the path of the ray in the substance. 

The object of the present section is to verify the laws of 
rotation of the plane of polarisation of light by solutions of 
certain substances. In these cases the rotation is very nearly 
proportional to the mass of dissolved substance per c.c. of the 
solution. The rotation produced by a layer 1 cm. thick of a 
solution containing 1 gram per c.c. is called the specific rotatory 
power of the substance. 

If a solution contains in 1 c.c. a grams of a substance the 
specific rotatory power of which is p, and a beam of plane 
polarised light passes through I cms. of the solution, and R is 
the rotation of the plane of polarisation produced, then 

R = a.l .p. 

Take about 85 c.c. of water, and add to it 10 grams of the 
sample of sugar provided. When the sugar has dissolved dilute 
the solution till its volume = 100 c.c. 

If the solution is coloured, mix with it about 2 grams of 
bone black, allow it to stand a few minutes and then filter. If 
the filtrate is then bright and clear it may be used. 

Arrange the solution tube provided, and two mounted 
Nicol's prisms, in such a way that the light of a Bunsen 



XLVII 



ROTATION OF PLANE OF POLARISATION. I 



225 



flame containing a bead of a sodium salt can be seen through 
the tube and prisms. 

The Nicol without the circular scale is to be placed between 




Fig. 97. 

the flame and one end of the tube, i.e. used as the " polariser," 
and that with the index and circular scale graduated in degrees, 




[^M 




Fig. 98. 

placed between the other end of the tube and the eye, i.e, used 
as the " analyser," Figs. 97 and 98. 

A thin sheet of yellow glass may be placed between the 
flame and polariser to cut off the blue light of the flame, and 
if necessary^ a lens may be used between the analyser and 
the eye. 

Wash out the tube thoroughly, and after filling it with 
distilled water place it between the prisms and rotate the 

s. p. 15 



226 LIGHT XLVII 

analyser till no light passes through the system. This will 
be the case when two lines similarly placed in the Nicols and 
at right angles to the line of sight are also at right angles to 
each other. Read the circular scale on the analyser. 

Repeat the observation several times, approaching the point 
of extinction from opposite sides each time, and take a mean of 
the results. 

Dry the tube by passing through it a plug of cotton-wool, 
and then fill it with the sugar solution, place it between the 
prisms, and determine as before the position of the analyser 
when no light passes. Repeat, and take the mean. 

The difference between the two means is the rotation 
produced by the solution. 

To test the truth of the law expressed by the equation 
R = a.l.p, take 20 c.c. of the solution, and dilute to 40 c.c. Fill 
a tube with this solution, and determine the rotation pro- 
duced. It should be half the amount previously obtained. 
Wash out another tube, fill with the solution, and place 
it between the prisms so that the light now passes through 
the two tubes in succession. Determine the reading for 
extinction. The rotation produced by both tubes will be 
found to be double that produced by one, and equal to 
that produced by one tube of double the strength. 

Arrange your results as follows : — 

17 Oct., 1895. 

Reading for darkness, with water tube ... ... 167 '2° 

„ „ with original solution 156*4° 

Rotation for original solution ... ... = 10*8° 

Reading for darkness, with solution of half strength = 161 '7" 

Rotation for solution of half strength ... = 5*5'' 
Reading for darkness, with 2 tubes of solution of 

half strength = 156*r 

Rotation for double length of solution of half 

strength ... ... ... ... ... = 11*1^ 



SECTION XLVIIL 

ROTATION OF PLANE OF POLARISATION. II. 

Apparatus required: Laurent or other polamineter, sugar, 
and sodium jlaine. 

With the apparatus used in the previous exercise the eye 
has been called upon to judge the point at which the minimum 
light passed through the Nicol's prisms. As the estimation 
of the exact position for a minimum is difficult, several instru- 
ments have been devised for getting over the difficulty, one of 
these being the Laurent polarimeter provided. By their means 
it is possible to estimate the position of extinction to a small 
fraction of a degi-ee. 

The Laurent instrument is arranged as shewn in Figs. 99, 
100. 




ttt^ 




Fig. 99. 

The light from the Bunsen burner A, in which a bead of a 
mixture of equal parts of sodium diborate and common salt 
is placed, passes through the lens li, the small hole in the 
diaphragm C, which is covered by a thin plate of bichromate 
of potash to cut off all but the yellow light, and falls on the 
lens and Nicol prism D. The plane polarised light which 
emerges from the Nicol falls on a plate of quartz E, which 
covers half the field. The quartz plate is cut so that the 

15—2 



228 LIGHT XLVIII 

optical axis is parallel to the edge which bisects the field. The 
plane polarised light falling on the plate is decomposed into 
two rays, one polarised in a plane parallel to the edge of the 
plate, the other in a plane perpendicular to this edge. The 
two rays traverse the plate with different velocities, and the 
thickness of the plate is so arranged that a difference of phase 
of half a wave-length is produced. The effect of this is, that 
if the light passing through the uncovered half of the field 
is polarised, say in the direction CA inclined at an , 
angle 6 to CB the edge of the quartz plate, then that 
which has passed through the plate is polarised in 
the direction CA' such that BCA' = BOA. On looking 
through the analysing Nicol K of the eyepiece, the two 
halves of the field will appear unequally bright, unless 
the principal plane of the analysing Nicol makes equal ^^' 
angles with the directions GA, GA\ i.e., is either parallel or 
perpendicular to GB. If it is parallel to GB the halves are 
equally bright, if perpendicular equally dark. The dark position 
is the one made use of, and the instrument is more sensitive 
the smaller 6, is consistently with sufficient light passing 
through the apparatus. Generally 6 does not exceed 2°. 

Adjust the eyepiece till the dividing line between the fields 
is seen distinctly when one half of the field is dark and the 
other light. 

The Nicol D is connected to a horizontal moveable arm, 
shewn in Fig. 100, and may be rotated within certain limits. Id 
order to get the position of greatest sensitiveness, determine 
first the position of the arm when the two halves of the field 
are equal in brightness whatever be the position of the Nicol K. 
This should be done with an empty tube at F. Now rotate 
D through a small angle not exceeding 2° by means of the 
moveable arm. It will then be found that the two halves of 
the field are equally dark for a certain position of K, the left- 
hand half increasing, the other decreasing, in brightness if K is 
moved in one direction, and the right-hand half increasing, the 
other decreasing, if the rotation is in the opposite direction. 

By means of the rotating screw (r, adjust the vernier to 
read 0, and by means of the tangent screw H, rotate K till 



XLVIII 



ROTATION OF PLANE OF POLARISATION. II 



229 



equality of fields is again produced. The instrument now 
reads when no active material is present. 

Insert at jF a tube filled with distilled water, and determine 




Fig. 100. 

the reading for equality of the fields, the adjustment being 
made alternately from opposite sides of the position of equality. 
This reading should still be if the water were pure, and the 
glass ends of the tube unstrained. 

Now insert a tube filled with a sugar solution containing 
10 grams of sugar per 100 c.c, and take readings several times 
as before. 

The sample of sugar from which the solution has been 
derived" may contain both sucrose or cane sugar, and glucose 



230 . LIGHT XLVIII 

or invert sugar, and impurities which will be assumed to be 
inactive. The cane sugar rotates the plane of polarisation to 
the right, and the invert sugar to the left. The observed 
rotation will be due to the difference of these effects. 

To determine the amount of each constituent present, we 
make use of the fact that when cane sugar is heated gently 
with acid it is converted into invert sugar, so that the whole 
of the sugar then present in the solution is invert sugar. 

From the two observations of the rotation the amount of 
each constituent present can be calculated. 

Take 20 c.c. of the original solution of sugar in a flask, add 
to it 2 drops of strong hydrochloric acid, add water till the 
volume is 22 c.c, and warm gently for 10 minutes, keeping the 
temperature about 80° C. 

Cool the resulting invert sugar solution by placing the flask 
in water, and after its temperature has fallen below 20° C. note 
the temperature, insert the solution in the tube, and determine 
the rotation produced. 

Let Ri be the rotation observed with the original sugar 
solution, and let R^ be that observed with the invert sugar, 
rotation to the right being considered positive and to the left 
negative. Let pi be the specific rotatory power of cane, p2 that 
of invert sugar, and let a^ grams of cane sugar and a., grams 
of invert sugar be present in 1 gram of the sample. 

Then in the first experiment, 

T^ 10a, -_ 10a., ^^ 

= 2/3iai + 2p2 ^2 (!)» 

and in the second experiment, 

J. _ JlOa, 360 lOaJ 20 
^"^^tl00'342^100J 22"^^' 

The factor 360/342 being due to the change from CiaHasC^n 
to C12H24O12 in inversion of the sugar, and the factor 20/22 due 
to dilution with acid. Reducing we have 



l'lR, = 2pAai-^-\-a} (2). 



XLVIII ROTATION OF PLANE OF POLARISATION. II 231 

Subtracting (2) from (1) we have 

R, - IIR, = {2p, - 2105/33) a,. 
Hence 

a, = ^ — _- __ , and a.^ = - - = — „ — / T^* 

2p, - 2-105/5, ' p, 2p^{px-p2) 

The quantity pi has been found to be nearly independent of 
temperature. Its value is + 6'65 degi-ees. 

/3o is negative and depends on the temperature t. Its value 
is - (2-78 - -03210 degrees. 

Record observations and results as follows : — 

15 Jan. 1896. 

Solution containing 10 grams of sample per 100 c.c. 
Reading with water in tube ... O'lO' to right. 

Do. sugar solution ) og^, 

12^24', 12^30', 12°30', 12^^24' ) "'^^''' ^iL. " " 

Rotation at temp. 15° C 12^7' „ „ 

Readine: with invert solution ) ^oi/x/ . i r^. 

3° 5(y, 4^30'. 4°5', 4^5' } "^^^°' _^ '" ^^^'• 

Rotation at 14° 5' C 4° 20^ „ „ 

Value of p2 M » ••• ••• =— 2°-31 

i2, = 12°-28, 122 = -4"-33. 

_ 12-28- 11 (- 4-33 ) _ 1714 
Hence a, - ^^.^^ _ g.^^. ^_ ^gi) - 18-15 " ^*' 

614-6-25 ^. 

"^= -2-3r-*^"- 

Or the sample contains 94% of cane sugar and 5^0 of 
invert sugar. 

Wash out the tubes thoroughly with tap water before putting 
away the apparatus. 



BOOK VI. 

MAGNETISM AND ELECTEICITY. 



SECTION XLIX. 



THE MEASUREMENT OF MAGNETIC FIELDS. 

Apparatus required : Mirror magnetometer, lamp and 
scale, bar magnet, and vibration box. 

The intensity of a magnetic field at a point is measured 
numerically by the force on a unit pole placed at that point. 
It can therefore be represented by a straight line, the direction 
of the line representing the direction, and the length representing 
the intensity, of the force. If two fields are superposed on each 
other the intensity of the combined field may be obtained in 
the same way as the resultant of two forces. If OP, Fig. 101 a, 
represents the direction and magnitude of the magnetic force 
on a unit pole placed at 0, due to one of the fields, and OQ 
that due to the other, the combined effect will be represented 
by the diagonal OR of the parallelogram OPQR. The magni- 
tude and direction of the resultant field may be calculated like 
the corresponding quantities in the case of a resultant force, 
by the following equations: 

OR' =^ OF' +0Q' + 20P.0Q cos POQ (1), 

OP ^ OQ ^ OR 
ain QOR sin POR sin POQ ^ ^' 



XLIX 



THE MEASUREMENT OF MAGNETIC FIELDS 



233 



Two special cases are of frequent occurrence. If OP and 
OQ are at right angles to each other (Fig. 101 6), the magnitude 
of the resultant is given by 

OR^ =0P'-\-0(^ 
and 

tan QOiJ=^ (3). 




Fig. 101 o. 




Fig. 1016. 



The second case (Fig. 101 c) is that in which the resultant 
OR is at right angles to one of the forces OP. Then 

OR* = OQ* - OP*, 
and 

sin Q0ie = 5|. 

Exercise. — It is found that the magnetic force on unit pole 
at a point distant r cms. from a straight wire of infinite length 
through which a current of strength i units is passing, is equal 
to 2i7r, and the lines of force are circles having the wire as axis. 
If the wire is vertical and carries a current of one unit, find by 
calculation or geometrical construction the direction in which 
a small magnetic needle will point when placed respectively 
east, west, north, south, and north-east of the wire at distances 
of 4, 11, or 15 centimetres from the wire. The earth's hori- 
zontal magnetic force may be taken as '17. 

If the magnetic field is due to a magnet having its north 
pole of strength fi Q,t B and its south pole of strength — fi at A, 
the magnetic force on unit pole at a point P due to the north 
pole at B is fi/BP', that due to the south pole at A is — /it/ilP*. 
If the point P is on the straight line through the poles A and 
B, the two forces act in the same straight line, and have a 



234 MAGNETISM AND ELECTRICITY XLIX 

resultant in the direction of the stronger, of intensity / such 
that 

T- ^ J_ _ 1_^- AB{BF-vAF) 
'^\BP' AP") ^ BP'AP' 

If we call the magnetic moment fjuAB of the magnet M, the 
distance of P from the centre of AB d, and if I is half the 
distance between the poles, i.e. about -f^ the length of the 
magnet, the resultant intensity 

2Md ,. 2d 

= M 



^d-iy{d + iy {d^-pf 

If the point P is so far from the magnet that I is small 

2 

compared to d, we have simply I = M.-^^. 

If the axis of the magnet is at right angles to the magnetic 
meridian, and H is the earth's horizontal force at the point P, 
the angle which the resultant force makes with the magnetic 
meridian is such that 

tan ^ = Tr > 

£1 

H- Yd—'^''^ W- 

A small magnetic needle placed at the point at which the 
resultant has been calculated will set itself in a direction 
forming an angle 6 with the magnetic meridian, and therefore 
enable 6 to be found. 

The resultant magnetic force at a point P due to a magnet 
AB, which in the above case has been calculated when P is on 
AB produced, may also be easily determined when P lies on a 
line through the centre C of the magnet perpendicular to AB. 

Since AP = BP the magnetic forces due to the two poles 
are equal, and their resultant bisects the angle between their 
directions, i.e. is parallel to AB. Each force having an in- 
tensity -jy^ has a component . ^g parallel to AB, hence the 

mtensity /= ^pa • 

Writing again M for the magnetic moment of the magnet, 



XLIX THE MEASUREMENT OF MAGNETIC FIELDS 235 

/ for half the distance between the poles, and d for the distance 

of P from C, we have I ~ r^ ;,T| • If P is so far from C that 

M 
I is small compared to rf, this becomes / = ^ . 

Hence the field at a distance d from the centre of a 
short magnet along the axis of the magnet, is twice the field 
at the same distance along a line through the centre of the 
magnet perpendicular to the axis. If the law of force had 
been the inverse ?jth power instead of the inverse square, the 
former force would have been found to be n times the latter. 
Hence an experimental determination of the ratio of these 
forces will furnish a proof of the law of action of magnetic 
poles on each other. 

If the axis of the magnet is at right angles to the magnetic 
meridian, and H is again the earth's horizontal force, the 
resultant of the two fields will make an angle 6 with the 
meridian where 

tan 6 = ^. 

or ^=((^-^ + Z-')Uan6> (5). 

The angle 6 may as before be determined by ])lacing a 
small magnetic needle at the point P, and observing the 
deflection produced when the magnet is placed in position. 

The apparatus provided consists of a small magnetic needle 
to which a mirror is attached, suspended by a fine fibre the 
torsion of which may be neglected. The centre of the needle 
is situated vertically over the middle points of the two 
horizontal graduated scales placed at right angles to each 
other, one of them being in the magnetic meridian. For con- 
venience the latter should be about 2 mms. below the former. 
The rotation of the mirror is determined in the usual manner 
by the motion of the image of a cross wire formed after reflec- 
tion by the mirror, on a scale placed parallel to the mirror when 
in its central position. The cross wire is attached to the scale 
and illuminated by a lamp. (For the method of obtaining the 
angles of rotation see Section XXXIV.) 



236 



MAGNETISM AND ELECTRICITY 



XLIX 



The deflecting magnet provided is first placed on the scale 
running east and west, say to the west of the needle, with 
its north pole towards the needle, the positions of the ends of 
the magnet are observed, and the scale reading of the cross wire 
is taken. The magnet is then reversed so that its south pole 
now points towards the needle, and the deflection again ob- 
served. It is then transferred to the east of the needle and 
the two deflections again observed. 

To make the corresponding observations with the magnet 
north and south of the needle, the magnet is placed on a frame 
which slides along the scale in the magnetic meridian, the 
magnet itself being at right angles to the meridian. The 
height of the frame is such that the magnet itself will be at 
the same level as when placed on the other scale. The frame 
is moved along the scale till the centre of the magnetic axis 
is at the same distance from the centre of the needle as pre- 
viously, the north pole of the magnet being say to the east. 
The deflection of the cross wire is observed, the magnet is 
reversed so that its north pole points to the west, and the obser- 
vation repeated. The frame and magnet are then transferred 
to the other side of the needle and the observations repeated. 

A set of observations should be taken for each of three 
distances of the magnet from the mirror as far apart as possible, 
and each set recorded as follows : — 

18 March, 1895. 
Determination of MjH at station B. 
Deflecting magnet A, length = 6"0 cms. .'. Z = 2*5 cms. 
Magnetometer 5, distance of scale from mirror = 103 cms. 



Position of Magnet 


Magnetometer 


mean 


tan 2d 


tand 


N. end 
cms. 


8. end 
cms. 


Middle 
cms. 


Beading 
cms. 


Deflection 
cms. 


P( 

250 W 
31-0 W 
31 -OE 
25 E 


)sition of r< 

310 W 
250 W 
250 E 
31-0 E 


3St 

28-0 W 
28-0 W 
28-0 E 
28-0 E 


500 

68-5 
31-6 
69-3 
30-2 


zero 

18-5 
18-4 
19-3 
19-8 


19-0 


•184 


•092 



XLIX 



THE MEASUREMENT OF MAGNETIC FIELDS 



237 



Similarly for the north and south positions. 
For the E. and W. positions in the above example we have:- 
M 



H 2d ob 



996. 



Similarly for the other positions. 

In order to determine both M and H it is necessary to find 
the value of some combination of the two other than the 
quotient, and the most convenient combination to detemiine 
experimentally is the product MHy which is connected with 
the time of torsional oscillation r of the magnet about a 
vertical axis through its centre of gravity, by the equation 



27r 



y MB 



where / is the moment of inertia of the magnet about the axis 
of oscillation. 

47r2/ 



Hence 



MH^ 



To determine the product MH, place the magnet which has 
been used as the deflecting magnet, in a light stirrup supported 
by a long thin fibre the toi-sion of which can be neglected, so 
that it can oscillate in a horizontal plane about a vertical axis. 
To protect the magnet from air currents suspend it in a box. 
Observe the times at which one end of the magnet passes a 
fixed mark on the bottom of the box in one direction, for six 
consecutive passages, the arcs of vibration not exceeding 20*". 
Wait a time nearly equal to that between the first and last 
observations, and then take six more observations. Arrange as 
follows : — 



Passage 
No. 


First Set ' 


Passage 
No7 


Second Set 


Time of 
10 oscillations 





Ih. 3m. 28. 


10 


Ih. 4m. lis. 


69 8008. 


1 


9 


11 


17 


68 „ 


2 


16 


12 


24 


68 „ 


3 


23 


13 


30 


67 ,. 


4 


29 


14 


38 


69 „ 


5 


36 ; 


16 


43 


67 „ 



Mean 

Time of one oscillation t 



»68 
= 6-8 



MAGNETISM AND ELECTRICITY XLIX 

Determine the moment of inertia of the bar magnet by 
weighing it to '01 gram, and measuring its length and breadth. 
The moment of inertia /, if m is the mass of the magnet, 2a its 
length, 26 its breadth, is given by the equation 

In order to determine whether a magnet is weak or strong, 
it is advisable also to calculate the magnetic moment per 
unit mass. 

Arrange observations and calculations as follows : — 

Mass of magnet ll'Ol grms. 
a= 3-0 cms. a2= 9-0 
h= -23 „ 6^= -05 



a^ + b^= 905 

.-. 7 = 1101 X 302 = 33-2 

, ,,^ 4 X 9-87 X 33-2 „„ . 
and MH = -r^-^ = 284. 

M 
Having obtained -^ and MH, M and H are determined as 

follows : — 



M = \/{^ {MH) = V996 X 28-4 = V2'8300 = 168. 




-1 £»0 

Magnetic moment of magnet per gram = ,- = 15*2. 



SECTION L. 

MAGNETIC SURVEY OF THE LABORATORY. 

Apparatus required : Magnet, vibration boXy and watch. 

If magnetic observations are taken in a room not specially 
constructed for magnetic work, it is quite possible that iron 
beams or iron pipes about the room may have an appreciable 
influence on the results. It becomes advisable under these 
circumstances to make a magnetic survey of the room, that is 
to determine the intensity and direction of the earth's magnetic 
force at a considerable number of points in it. 

As change of direction of the horizontal magnetic force is in 
most cases less objectionable than change of intensity, attention 
should be given chiefly to the latter. It will not be necessary 
to measure the horizontal force at different places, but only to 
compare them with each other. This is done in the simplest 
way by measuring the times of vibration of a magnetic needle 
ab<jut a vertical axis of suspension, as the horizontal forces will 
be inversely proportional to the squares of these times. 

The apparatus supplied is provided with a horizontal scale of 
degrees which enables the direction of the magnetic force, i.e. of 
the magnetic meridian, to be detennined. A magnetic needle 
to which a pointer moving over the scale is attached, is sus- 
pended from a rod by means of a fibre which should be free 
from torsion. To secure this, take out the needle and suspending 
fibre, and holding the needle in the hand allow the suspending 
rod to hang down until the fibre is completely untwisted. 

The direction and magnitude of the earth's magnetic force 
are to be determined at each of the places shewn in the plan of 
the laboratory*. 

* A plan of the laboratory with the positions at which observationn are to be 
taken marked on it, ihoald be taapended in the laboratory. 



240 



MAGNETISM AND ELECTRICITY 



At each place the box within which the magnet swings, is to 
be placed with its long sides, and therefore the zero line of the 
scale, parallel to the outer wall of the laboratory. The needle 
should then be set into oscillation about the axis of suspension, 
and the readings of one end at five successive turning points, three 
on one side of the position of rest and two on the other, taken. 
The mean of the three on one side, and the mean of the two on 
the other, should then be found, and the mean of these means 
taken. Repeat, reading now the other end of the needle. The 
mean of the results for the two ends is the angle between the 
magnetic meridian and the outer wall of the laboratory. 

To measure the time of oscillation with sufficient accuracy 
for our purpose, note down to the nearest second as in the 
example given below, the times at which the needle passes 
through its position of equilibrium in the same direction, 
during say six successive swings not greater than 20°. 

Then wait a time approximately equal to that between the 
first and last observations and take another set of six observa- 
tions of the times of passage. 

Two columns may now be formed by entering the observed 
times as follows : — 



OsciUation 
No. 


First Set 


Oscillation 
No. 


Second Set 


Time of 10 
oscillations 




1 

2 
3 

4 
5 


11 h. 23 m. 3 sec. ' 

13 „ ; 
17 „ I 
21 „ i 

25 „ 1 


10 
11 
12 
13 
14 
15 


23 m. 49 sec. 

53 „ 
59 „ 

24 m. 4 „ 

9 » 
13 „ 


46 sees. 

45 „ 

46 „ 

47 „ 

48 „ 
48 „ 








Mean = 


46-7 „ 



Time of 1 oscillatioD = 4*67 „ 



The differences in time between corresponding passages in 
the first and second sets are entered in the third column and the 
mean taken. This mean should be correct to half a second, i.e. 
the error should not exceed 1 per cent. 



MAGNETIC SURVEY OF THE LABORATORY 



241 



In order to make certain that no mistake in the number 
of oscillations has been made, the experiment should now be 
repeated. 

If during your experiments there are any movable masses 
of iron within a metre of the magnetic needle, record the fact 
and state their positions. 

The horizontal force being inversely proportional to the 
square of the time of oscillation, we may obtain numbers pro- 
portional to the horizontal force by calculating the reciprocals 
of the squares of those times. (Barlow's Tables will save much 
arithmetical work.) 

Taking the given values of H and of the magnetic declina- 
tion for the point of reference marked in the plan, determine 
the intensity and direction at the other points. 

Enter results as follows, giving also the plan with the 
points of observation marked on it. 

26 March, 1900. 







1 




Angle to 


Angle to 




Place of 
observation 


T 


fi 


H 


Laboratory 
Walls 

19° E 


Geographical 
Meridian 


Remarks 


1 


11-61 


•00741 


-192 


3°W 




2 


10-61 


885 


-221 


8"E 


14° W^ 




3 


14-61 


467 


-126 


69° E 


47° E 1 


much 


4 


8-77 


-01299 


-323 


21° E 


rw 


disturbed 


6 


10-81 


855 


•221 


9°E 


13° W J 




6 


12-38 


653 


•168 


8°E 


14°W 




7 


12-27 


662 


•172 


3° W 


25° W 




8 


12-33 


•00657 


•170 


5°E 


17° W 


reference 
point 



■.P. 



16 



SECTION LI. 

THE TANGENT GALVANOMETER AND OTHER 
CURRENT METERS. 

If a wire is bent into the form of a circle of radius r, and a 
current of strength i c.g.s. units is sent through it, a magnetic 
field is produced, which, at the centre of the circle, and at 
distances from the centre not greater than '08 of the radius, 
is sensibly uniform, of strength F = ^irilr, and in direction 
perpendicular to the plane of the wire. Taking the earth's 
magnetic field into consideration, the total field at the centre 
of the circle is the resultant of the fields due to the earth and 
the current in the wire respectively, and a magnetic needle 
supported at its centre will set itself along the direction of the 
resultant field. If the plane of the wire is coincident with the 
magnetic meridian, the component forces are at right angles, 
and if 6 be the inclination of the resultant to the magnetic 
meridian, and H the earth's horizontal force, we have 

tan = -rj. 

Hence, substituting the value of F in terms of i, we have 

rH 

i= TT- tan 0, 

ZTT 

which will enable us to determine i ii H and r are known, and 
6 is observed. 

From the presence of the tangent in this expression, this 
arrangement of a circular coil of wire with a magnetic needle at 
its centre is called a Tangent Galvanometer. 



LI CURRENT METERS 243 

In order to make the instrument more sensitive, the current 
to be measured may be sent through a wire coiled n times 
round the ueedle. In that case the magnetic field produced by 
the current is n times that produced by a single wire of radius 
equal to the mean of the radii of the n turns, and we have 

*=£*-^ w- 

The practical unit in which currents are measured is the 
Ampere, which is one-tenth of a c.g.s. unit, hence, if we call A 
the number of amperes passing through the coil, we have 

^ = ^l^tan(9 (2). 

riTT 

Since 5r/mr depends only on the coils of the galvanometer, 
we may put it equal to l/G and call G the " coil constant of the 
galvanometer," and we have 

A = ^ tan 0. 

(jr 

If the galvanometer is used in a fixed position, H will 
be constant, and we may write H/G = k. We then have 

A =k tan 0, 

where k is the current which produces a deflection of 45°, and 
is generally called the " current constant of the galvanometer." 

It is, however, unsafe to assume that H is uniform through- 
out buildings, owing to the irregularity of the distribution of 
the earth's line of force, produced by masses of iron in the 
walls and floors. 

Hence, if a current is to be measured in amperes by means 
of a tangent galvanometer, the horizontal component of the 
earth's magnetic field at the point of observation must be 
determined. This is best done by taking the times of oscil- 
lation of a magnet about a vertical axis, at the point of 
observation, and at some point at which the value of H is 
known. (See p. 241 Magnetic Survey, or Intermediate Practical 
Physics, p. 184.) 

In order to form a judgment of the power of the instru- 
ment, we must estimate the accuracy with which an observation 

16—2 



244 MAGNETISM AND ELECTRICITY LI 

of the deflection may be taken, and calculate from this the 
en-or which may be introduced in the result. If the circle of 
the galvanometer is properly graduated, we ought to be able to 
read the deflections to '1 degree, but it would be difficult to 
secure a much greater accuracy. An error of "1 division will 
according to the investigation (p. 6) affect the determination 
of the current much more in some parts of the scale than in 
others. 

Numerical Exercise. Calculate the percentage error 
introduced in measuring a current by an error of '1 degree 
in the reading of a tangent galvanometer, the deflections 
being 10°, 20°, 30°, and so on to 80°. 

When the deflection of a galvanometer needle changes, its 
poles trace out a circle, and in order that the tangent law may 
be accurate, the magnetic forces F and H should be the same 
at all points of the circumference of this circle. We require, 
therefore, to know how long the magnetic needle may be 
without the deviations from the tangent law introduced 
by variations of F exceeding the errors introduced in other 
ways. It may be shewn that the greatest error due to the 
finite length of the needle is introduced when the deflection 
is 60°, that it is proportional to the square of the length, and 
that if the ratio of the length of the needle to the diameter of 
the coil is less than 08, the error introduced by the want of 
uniformity of the field is never greater than that caused by an 
error of '1° in the reading of the deflection. 

Equation (2) shews that the sensitiveness of a tangent 
galvanometer may be increased either by diminishing the 
radius r, or by diminishing H, or by increasing the number of 
turns n. 

If the radius of the coil is too much diminished, the tangent 
law ceases to hold unless the needle is also shortened. 

The horizontal force H may be diminished and the sensi- 
tiveness in consequence increased by permanent magnets placed 
.80 as to counteract the earth's magnetic force. 

Numerical Exercise. Design a tangent galvanometer of 
-one turn which, when placed in a field of strength '17, will give 
a deflection of 45° for a current of 1 ampere. 



LI CURRENT METERS 245 

Adjustment of the tangent galvanometer. A tangent galva- 
nometer before use should be adjusted so that the plane of the 
coil through which the current is to be sent, is parallel to the 
magnetic needle. The coil will then be in the magnetic meridian. 
As the deflected magnet is short, a pointer is attached to it, so 
that the rotations may be more easily read ofif on a circular 
scale. When the needle is in the plane of the galvanometer 
coil, the pointer should stand at the zero of the scale. As this 
condition may not be accurately satisfied, any error is eliminated 
by sending the current through the galvanometer first in one 
direction and then in the other, the deflections being noted 
each time. If the zero of the scale is correctly placed, the two 
deflections will be equal and opposite. If the adjustment is 
not quite right, one deflection will exceed the true deflection by 
the same amount that the other is in defect, and the mean of 
the two will give the true deflection. 

If the axis about which the needle rotates is not quite at 
the centre of the scale, small errors will be introduced which 
are, however, eliminated by reading the two ends of the pointer 
(see note p. 202). Each observation of a tangent galvanometer 
consists, therefore, of four readings, the mean of which gives as 
correct a value as is obtainable with the instrument. 

In a second type of current measuring instruments, known 
as electro-dynamometers, the magnetic needle of the galvano- 
meter is replaced by a coil of wire through which the current 
to be measured is sent. The passage of the current through 
the coil makes it behave as if it were a magnet with its axis 
along the axis of the coil. The coil will therefore set itself 
with its axis along the resultant field due to the earth and 
to the current in the fixed coil. In general the movable coil 
is suspended by means of the wires which carry the current 
to and from it, and the suspension introduces another force, 
tending to keep the coil in the position it occupies when no 
current passes, i.e. with its axis in the plane of the fixed coil. 
If the force introduced by the suspension greatly exceeds that 
due to the presence of the coil in the earth's field, the latter 
may be neglected, and a direct relation between the strength 
of the current in the two coils and the deflection obtained. 



246 MAGNETISM AND ELECTRICITY LI 

Instead, however, of allowiug the movable coil to be deflected, 
we may increase the force due to the suspension, by twisting 
the head carrying the upper ends of the suspending wires in a 
direction contrary to the deflection, till the deflection is reduced 
to zero. The magnetic force tending to deflect the movable 
coil, is then equal and opposite to the force on the coil intro- 
duced by the rotation of the torsion head. The magnetic force 
is proportional to the magnetic moment of the suspended coil, 
and to the strength of the field in which it is placed. It will 
therefore be proportional to the square of the current, since 
both the strength of the field in which the suspended coil 
moves, and the moment of the magnet to which the suspended 
coil is equivalent, are proportional to the current. 

In the Siemens electro-dynamometer the torsional force is 
provided by a spiral spring, to the top of which a pointer 
moving over a dial is attached. This pointer is rotated by 
means of the torsion head till a second pointer attached to the 
moving coil is at the zero of the scale. The planes of the two 
coils are then at right angles to each other, and the torsional 
force exactly balances the magnetic force. The torsional force 
being proportional to the angle 6 of twist of the torsion head, 
we have C^ x 0, or C = K\/d, where K is the constant of the 
instrument. 

In order that the deflecting effect of the earth's field on the 
suspended coil may be least, that coil should be placed with its 
plane perpendicular to the magnetic meridian. 

Another form of instrument depending on the action of 
currents on currents, is the Kelvin current balance, where two 
horizontal coils are suspended from the arms of a balance, and 
each is acted on by two similar coils placed above and below 
it respectively. The current to be measured is sent through 
all the coils, in such directions that the suspended coil on the 
left is pulled down and that on the right pulled up. This pull 
is balanced by a weight sliding along a scale attached to the 
arm of the balance, and the distance through which this weight 
has to be moved to produce equilibrium, will be proportional to 
the square of the current passing through the coils. 

Another type of galvanometer which has recently come 



LI CURRENT METERS 247 

much into use is the D'Arsonval. It consists of a movable coil 
suspended in a magnetic field, the lines of force of which are 
parallel to the plane of the coil. When a current is sent 
through the coil, it tends to set itself so that the greatest 
number of lines of force pass through it, and a deflection 
results which can be read off on a scale, a beam of light being 
reflected from a mirror fixed to the coil. The magnetic field of 
the instrument being many times stronger than the earth's 
field, its indications are independent of magnetic disturbances, 
such as occur in the neighbourhood of electric railways. 

If the current an instrument is intended to measure is very 
small, the coils consist of a great many turns of fine wire, and 
if the resistance of the instrument, which is in consequence 
high, is constant, the indications may be taken as measures of 
the electromotive forces or differences of potential applied to 
the terminals, i.e. the instrument may be used as a "voltmeter." 



SECTION LIL 

COMPARISON OF CURRENT METERS. 

Apparatus required : A small storage cell, a tangent 
galvanometer arranged so that one, two, or three turns of wire 
can he used, the constant with the three turns being about 1, a 
Siemens Dynamometer, having a constant about % a set of 
resistances each about 'o ohm which can be joined up in different 
ways so as to alter the resistance in circuit, and a commutator to 
enable the current to be reversed. 




Fig. 102. 



B 



\1M 



Si 



@ 



R K 



Fig. 102 a. 



When instruments intended for the same purpose are con- 
structed on different lines, it is advisable to compare their 
indications together, and the present exercise consists in com- 
paring the indications of the tangent galvanometer with those 
of an instrument such as the electro-dynamometer or Kelvin's 
" ampere balance," constructed according to a different principle. 



LI I COMPARISON OF CURRENT METERS 249 

Connect the storage cell B (Figs. 102, 102 a) through the 
resistance coils R to two opposite quadrants of the four-way 
key Ky the plugs being removed. Place the electro-dynamo- 
meter S with the plane of its fixed coil in the meridian, at 
least 20 cms. from the resistance coils, and the tangent galva- 
nometer G with its coil also in the meridian, a metre away 
from the dynamometer. Connect them in series with each 
other to the other terminals of if as shewn. See that the 
galvanometer needle swings freely, then level the dynamometer, 
and screw down the spring on the base till the movable coil 
is released. Rotate the torsion head in the centre of the dial, 
and observe whether the coil moves freely. 

Arrange three of the '5 ohm resistance coils in series. 
Connect together the two wires leading to the tangent galva- 
nometer by placing both under the same binding screw on the 
base of the instrument, and make the circuit at K by inserting 
plugs in the two holes in one diameter of the key. While 
contact is being made, watch carefully the needle of the 
galvanometer in order to detect if there is any displacement 
due to a direct action of the current in the leads. If so, 
twist the leads between K, G, and S round each other. This 
probably will stop the action. If it does not, reverse the 
direction of the current by inserting the plugs in the holes 
in the other diameter of the key and see whether the effect 
is reversed ; if not, the cause is to be found in a direct effect 
of the current in the coils R on the galvanometer needle. 
Change the position of R till this effect disappears. All extra- 
neous effects of the current on the magnetic needle of G are thus 
got rid of before the galvanometer is used as a measuring instru- 
ment. In all future experiments students will be expected to 
avoid errors due to the magnetic effects of the leads on the 
measuring instruments, without being specially directed to do so. 

Connect the wires leading to the galvanometer to the two 
terminals of the instrument between which 1 T {i.e. one turn) 
is marked, and complete the circuit at the plug key. A 
deflection will be produced and the needle will take some time 
to come to rest. Students should therefore practise making 
connections in such a way that the needle is not set into violent 



250 MAGNETISM AND ELECTRICITY LII 

oscillations, introducing for the purpose an extra plug key into 
the circuit close to the galvanometer. The following rules will 
be found useful : — If the needle is at rest in its normal position, 
send the current through the galvanometer for ^th of the com- 
plete time of oscillation of the needle, then break the circuit. 
The needle will continue to move but with diminishing velocity. 
When it is on the point of turning back, make contact again, 
this time permanently. If the needle moved without friction, 
it would be brought to absolute rest in its new position 
of equilibrium. During the first period the needle moves 
through half the ultimate angle of deflection, hence if the new 
position of equilibrium is approximately known, it will be 
easier to fix the end of the period by the position of the 
needle than by an estimate of time. Thus, if it is known for 
example, that the needle will be deflected through about 40°, 
make contact until the needle passes through 20°, then break 
circuit. The needle will continue to move owing to its inertia 
till it reaches 40°. As soon as it begins to turn back, make 
contact permanently. 

If the needle is deflected by a current which is to be 
interrupted, break the circuit until the needle is half way 
between its original position and the zero point, make the 
circuit again until the needle comes to rest at the zero, then 
interrupt permanently. 

Students are recommended to practise these rules till they 
are able to avoid producing large oscillations of the needle and 
consequent loss of time. 

The student should prepare his note-book for recording 
the observations by ruling columns according to the scheme 
given further on. The headings of the columns should be 
written out completely before the experiments are com- 
menced, so as to leave nothing but the actual numbers to be 
entered. 

Break the circuit, adjust the torsion head of the dynamometer 
till the pointer attached to the coil reads zero, read to 1° the 
position of the torsion head pointer, and of the ends of the 
galvanometer needle and record as shewn below. 

Re-make the circuit, adjust the torsion head till the pointer 



LI I COMPARISON OF CURRENT METERS. 261 

reads zero, read to '1° the position of the torsion head pointer, 
then the position of each end of the needle of the galvano- 
meter, and finally again the dynamometer. Reverse the 
current and take the readings in the same order. In each 
set of observations there are, therefore, four readings to be 
recorded for each instrument. 

The galvanometer connections and resistances are to be 
varied as follows ; and observations taken as above : — 

I. Current passing through one turn of wire on the 

galvanometer (connect to terminals between which 

1 T is marked). 

(1) 3 Resistance coils in series. 

(2) 1 Resistance coil only. 

(3) 3 Resistance coils in parallel. 

II. Current passing through 2 turns of wire on the 

galvanometer (connect to terminals between which 

2 T is marked). 

The resistances to be varied as in I. 

III. Current passing through 3 turns of wire on the 
galvanometer (connect two previous sets in series). 
The resistances to be varied as in I. 

In order to compare the instruments, calculate the current 
indicated by the dynamometer from the formula A = Ksjd 
where d is the deflection and K is the constant of the instru- 
ment. Measure the outside diameter of the turns of ^vire of 
the galvanometer and the thickness of the wire. Calculate 
the mean diameter of the turns. Determine the value of the 
horizontal component of the earth's magnetic force at the point 
of observation, by taking the time of oscillation of a horizontal 
magnet at the point, and at one at which the value of the force 
is known. Calculate the current constant of the galvanometer. 

Determine the mean deflections of the galvanometer to 
two decimal places, and the currents by the formula il =7 tan 
to three figures, and tabulate as follows : — 



252 



MAGNETISM AND ELECTRICITY 



LII 



29 April, 1896. 

Comparison of Tangent Galvanometer (A) with 
Siemens Dynamometer (A). 



Outside diameter of turns of galvanometer... 
Diameter of wire, including insulation 
Mean diameter of turn 
Mean radius of turn (r) 

Value of earth's horizontal force = 17 ( jTry) 

= {H) 

Calculated constant of galvanometer : 1 turn 

2 turns 
.. * „ 3 turns 



= 19-86 cm. 
= -12 cm. 
= 19-74 cm. 
= 9-87 cm. 



•17. 

2-67. 

1-34. 

•89. 



Resist- 
ances 


1 

s 


Dynamometer. 
Constant=-15 


Galvanometer 1 turn. 
Constant =2 -67 


Ratio 


Read- 
ing 


Mean 
defltn 


v/def. 


Cd 


Readings 


Deflection 6 


tand 


•89 tan e 


OdIC^ 


Wend 


Eend 


Wend 


Eend 


Mean 






P-5 


zero 






5n 


l°-Os 


zero 












3 in 

series 


a 


^24-2 
i23-8 


22-5 


4-73 


•711 


15-5n 


lo^ls 


15-0 


14-1 


14-55 


•278 


•722 


•98 


it 


/S 


\25-2 
/24-8 


23-5 


4-84 


•727 


14-38 


15-ln 


14-8 


15-1 


14-95 


-286 


•740 


-98 


single 


iS 


i54-5 
J54-7 
























»> 


a 


^55-5 
J55-3 
























Bin 
parall. 


a 


^80 
J80-3 
























>» 


^ 


j78 
177-6 




























1-5 


zero 





















Make out similar tables for the observation with two and 
three turns of wire on the galvanometer. 



SECTION LIIL 



VERIFICATION AND APPLICATION OF OHM'S LAW. 



Apparatus required : Tangent Galvanometer having a 
constant about 1, or an ammeter, voltmeter reading to 2 volts, 
1 meter of No. 25 platinoid, No. 27 copper and No, 31 iron 
wire, 50 cms. of No. 29 platinoid vjire, all mounted on graduated 
boards, wire gauge. 

The law discovered by Ohm in 1827 states that the ratio of 
the difference of potential between two points of a conductor 
through which a current is flowing, to the current flowing, is a 
constant which depends only on the material, temperature, and 
shape of the conductor between the two points, and is indepen- 
dent of the strength of the current. It is the object of the 
present exercise to verify this law and to shew some of its 
applications. 




(if & 




^ 



Pig. 103. 



J). ..-^n 



^^m 



Fig. 103 a. 



Exercise 1. To verify that the difference of potential be- 
tween two points on a conductor is proportional to the current 
passing through the conductor. 

In order to verify this law, arrange (Figs. 103 and 103 a) a 
storage cell E in series with a four- way reversing key K with 
the plugs out, a tangent galvanometer or ammeter G, and a 
meter of No. 25 platinoid wire AB, stretched along a graduated 



254 



MAGNETISM AND ELECTRICITY 



LIII 



board. See that all connections are good, none of the wires 
being loose in the binding screws. Clean the movable knife- 
edges provided, place them on the platinoid wire at G and D 
20 cms. apart, and connect them to the terminals of the volt- 
meter or high resistance galvanometer F, capable of reading up 
to 2 volts. Head the description of the instrument and the 
instructions for its use supplied with it. 

See that the pointer swings freely and stands at zero when 
the circuit through the instrument is broken. 

Make the circuit at K, read the deflection first of V, then of 
G, and again of F, estimating to tenths of a division. Reverse 
the current at K and repeat the observations. 

Insert a thin wire of copper into the battery circuit, in 
series with the platinoid wire, and repeat the observations of 
the deflections, keeping the contacts on the platinoid wire at 
the same points. 

Introduce further a thin wire of iron into the circuit and 
repeat the observations. 

Introduce lastly the short length of No. 29 platinoid and 
repeat the observations. 

Break the circuit at K. 

If necessary reduce the readings of the instruments to 
quantities proportional to currents, and record as follows : 

12 September, 1897. 



Exp. 


Commu- 
tator 


Deflections 


Batios 


Voltmeter. 
Direct reading 


Tangent 
galvanometer 


Tangents 


1 
2 
3 
4 


a 
b 

b 
a 

a 
b 

b 
a 


It] «•« 




•96G 
•933 
•543 
•368 


26-7 
26-8 
26-2 
26-5 



The constancy of the ratio in the last column verifies the 
above statement. 



LIII 



VERIFICATION AND APPLICATION OF OHMS LAW 



255 



The constant ratio of the difference of potential between 
two points of a conductor, to the current which it produces in 
the conductor, is called the " Resistance " of the conductor 
between the two points. 

Exercise 2. To verify that the resistance of a homogeneous 
conductor of uniform cross section, is proportional to its length. 

With the cell circuit arranged as at the last observation, place 
one knife-edge on the platinoid wire at the division marked 10, 
and keep it there permanently during the experiment. Place the 
second knife-edge at the division marked 20, close the circuit 
at K, and keep it closed during the remainder of the experiment. 
Wait one or two minutes till the wires, which are being heated 
by the current, take up a stationary temperature, note the time, 
then take readings of both instruments. Move the knife-edge 
from 20, successively to 30, 40, 50, 60, 70, and 80, and then back 
again, reading the voltmeter at each position. Read the current, 
note the time, and record the observations as follows : 

12 September, 1897. 



Deflections direct 


Time of first reading : 1 h. 43 m. 




rime of last 


reading: 2h. Om. 


Galvanometer 20*2 




Galvanometer 20-1 






Deflection of 








Knife Knife 
edge edge 


Length 
cms 


Voltmeter 


Mean 
Deflection 


Difference 


Deflection 
length 


A at 


Bat 








Ist series 


2nd series 








10 


20 


10 


4-9 


4-8 


4-85 


4-85 


•485 




30 


20 


9-8 


9-8 


9'8 


4-95 


-490 




40 


30 


14-7 


14-6 


14-65 


4-85 


-488 




50 


40 


19-8 


195 


19-65 


500 


•491 




60 


60 


24-5 


24-9 


24-7 


505 


-494 




70 


60 


29-5 


29-6 


29-55 


4-65 


•493 




80 


70 


34-2 


34-2 


34-2 


4-65 


•489 

■ 1 



If the strength of the current has altered during the 
experiment, the first and second set of readings will not quite 
agree. But if the change in the current has been uniform, and 
if the second set of readings has been taken in the reverse 



256 



MAGNETISM AND ELECTRICITY 



LIII 



order to the first, the mean of the two sets should give the 
readings con-esponding to the time half-way between the 
beginning and the end of the experiment. 

The column headed differences in the table of results, 
contains the differences between each pair of successive numbers 
in the previous column. These differences, which correspond 
to equal differences in the length of wire CD, should be nearly 
equal. 

Small deviations from equality are produced by : 1st, errors 
in reading ; 2nd, want of uniformity in the wire AB ; 3rd, errors 
in the voltmeter scale. If there is a large deviation it is 
probably due to one of the contacts being bad. 

Reverse the commutator in the cell circuit and repeat the 
whole series of observations, entering them in a second table 
headed — Deflections reversed. 

Exercise 3. To verify that the resistance of a homogeneous 
wire varies inversely as the area of its cross section. 

Without disturbing the circuit through the tangent galvano- 
meter, place the knife-edges on the No. 29 platinoid wire, at a 
distance of 10 cms. apart, and observe the deflection of the 
voltmeter. Now place the knife-edges 10 cms. apart on the 
No. 25 platinoid wire and again observe the deflection. 

Measure the diameter of each wire in the middle of the 
length between the voltmeter contacts, and enter the deflec- 
tions, the diameters and areas of the cross sections of the wires, 
as follows : 

Galvanometer reading , _i20'l. 



Wire 


Commu- 
tator 


Voltmeter 
deflection 


Diameter of 
wire 


Cross 
section 


Voltmeter 
deflection 
X section 


No. 25 
No. 29 


a 
h 

h 
a 




•0504 cm. 
•0352 cm. 


•00020 
•000361 


•0266 
•0264 



The last column shews that the above statement is correct. 



LIII VERIFICATION AND APPLICATION OF OHM'S LAW 257 

Exercise 4. To find the relative specific resistances of 
copper, iron and platinoid. 

Keeping the main circuit undisturbed, place the knife- 
edges connected with the voltmeter on the copper wire 80 cms. 
apart, and take readings of both ammeter and voltmeter. 

Remove the knife-edges from the copper, place them 40 cms. 
apart on the iron wire, and read the voltmeter. Now place 
them 20 cms. apart on the No. 29 platinoid, and take readings. 
The ammeter is read in order to make sure that the current 
has not changed during the experiment and the following 
calculation is made on that supposition. Measure the diameters 
of the wires in the middle of the lengths used. From the 
observations calculate the deflections which would have been 
obtained with 1 cm. of each wire between the knife-edges. 
Then calculate what would have been the deflection if the 
cross sections of the wires of 1 cm. length had been 1 sq. cm. 
These deflections will be proportional to the specific resistances 
of the materials of the wires. 

Assuming the specific resistance of copper to be 1, calculate 
those of iron and of platinoid. 

Tabulate as follows : 

Jan. 5, 1900. 
Galvanometer Reading at beginning 20' 1 

.. end 201 



1 

wire 


Length 


Voltmeter 
deflection 


Deflection 
per cm. 


Diameter 


Cro« 
section 


DeflecUon 
perrai. 
xeect. 


BeUUTe 

apeciflc 

resistance 


Copper 27 
Iron 31 
PUtinoid 29 


80 cms. 
40 „ 


b 2-6i 2^ 


•0326 

•890 

•988 


•0417 
•0296 
•0862 


•001.14 

•000,682 

•000.961 


•000,0369 

■000,266 

•000.898 


1 

72 

248 



8. P. 



17 



SECTION LIV. 

ARRANGEMENT OF CELLS. 

Apparatus required: Tangent galvanometer with a con- 
stant of '2 or '3, reversing key, four Daniell cells, resistance 
coils, and connecting wires. 

The Tangent Galvanometer may be used to illustrate the 
different ways in which voltaic cells may be joined together, 
and to shew which way is the most advantageous under given 
circumstances, but a more sensitive instrument than that used 
in the previous exercises will be necessary. 

Set up four Daniell cells, having porous pots as nearly alike as 
possible. If the porous pots are dry, allow them to stand fully 
immersed for half-an-hour in a bath of zinc sulphate solution, 
in order to fill the pores of the pots with the solution. Remove 
them from the bath, and place each in a jar containing a zinc 
plate dipping into zinc sulphate solution. Then fill each 
porous pot with concentrated copper sulphate solution, and 
place a copper plate inside it. 

Place the galvanometer with its coil in the magnetic 
meridian and read the zero. 

In order to find the resistance of each of the four Daniell 
cells, connect one cell through a commutator or reversing key. 




Fig. 104. 



LIV 



ARRANGEMENT OF CELLS 



259 



and a coil having a resistance of R ohms (about 4), with the 
galvanometer, and observe the deflection, (1) with the re- 
sistance of R ohms, (2) without the resistance, and (3) once 
more with the resistance in circuit, two observations being 
taken in each case, once with the current in one direction, 
and once with the current reversed. Read again the zero. 
Calculate the mean deflections and their tangents. If f/ is 
the mean of the tangents in (1) and (3), and ti is the tangent 
in (2), we have 



Kt,^ 



B-i-G-hR 
E 



where K represents the galvanometer constant, E the 
electromotive force, B the resistance of the cell, and G the 
resistance of the galvanometer, which for the present purpose 
may be neglected, as it is small compared to B and R. 
Putting G = 0, we deduce from the two equations 

Proceed in this way to determine the resistance of each of 
the four cells. 

Enter your observations and results as follows : — 

23 Jan., 1894. Resistance of Daniell Cell 1. 



ReflisUnce 

in circuit 

ohms 


Readings 


Deflections 


Tangents 
of means 


Mean 
resistonce 


BMt West 


East 


West 


Mean 


4 01 j 
4-01 j 


•6ii 

35-5 n 
85-08 

48-0 8 
48-7 n 

85-5 ri 
86-08 

•5n 




85 2 8 
85-0 n 

48-3 n 
48-5 8 

85-08 
84 -80 




Zero 

36 n 
35-5 8 

48-5 8 
48-2 n 

85 -On 
85-58 

Zero 


35-2 8 
35 On 

48-3 n 
48-5 8 

85-08 
84 -8 n 


35-2 

48-3 

j 35-1 


-705 

l-122 = f, 
-703 


•704 ^f/ 



* The mean defleotion can be obtained by taking the mean of the four 
readings of the ende of the pointer without making uee of the zero, but it is 
advisable in this exercise to adopt the method shewn in the example, as it 
enables mistakes to be more easily detected and corrected. 

17—2 



260 MAGNETISM AND ELECTRICITY LIV 

•704 
.-. Resistance B^ of cell 1 = 401 ^ =675 ohms. 

Make out a similar table for each of the four cells. 



Connect all the cells in series through 8 ohms external 
resistance (Fig. 104), to the galvanometer, observe the de- 
flections and find the tangent of the mean angle of deflection 
produced by the current in the two directions. 

From the observed deflections with the separate cells, 
calculate the deflection which ought to have been obtained, 
as follows: — 

For the first cell we have — 

l~/{" 

and similar equations for the others. Hence 



i.e. if 



\ I I I ~ K 

- = -(- - 1 -^ 



t will be the tangent of the angle of deflection when the cells 
are all connected in series to the galvanometer. Hence if t' is 
the tangent of the angle of deflection with 8 ohms in series, 

/ _ Bi-\- B^-\- B^-)r B^ 
~ B^ + B^ + B, + B,-\-S' ' 

Calculated If = '649. 

Observed t' = '640. 



Arrange all the cells in parallel, and observe the deflections 
with 4 ohms in series with them. Calculate the tangent of 
the mean deflection. 



LIV 



ARRANGEMENT OF CELLS 



261 



Also calculate 


the deflection from 


the 


observations made 


with the separate 


cells, as follows: — 






For the first cell we have 








U E 
1 ~ K 








b; 






and similarly for the other cells. Hence 






t^ + f^-hU-h u 


E 








K' 





1.6. if t^ti + ti + ts + ti, t would be the tangent of the deflection 
if the cells were connected in parallel with the galvanometer. 
Hence if t' is the tangent required, for the case that the 4 ohms 
are inserted 

t 



t' = 



1 



1 



«) 



+ 1 



\B, ' B^' B, 
Calculated ^'=1118. 
Observed ^'=1-120. 



Arrange the cells in two parallel sets of two cells in series, 
and determine the tangent of the deflection with 4 ohms in 
senes with the galvanometer. 

Establish a formula and calculate the deflection similarly. 



Determine by experiment which of the three arrangements 
of the cells previously tried, gives the greatest current through 
external resistances of 1, 4 and 8 ohms respectively. 

Tabulate the results as follows, underlining the maximum 
deflection in each 



External 


Mean deflections 


Cells in 
parallel 


In two sets 
in parallel 


In aeries 


1 ohms. 


65-3' 

481 

33-7 


39-2' 

30-2 

21-6 


42* 

36-9 

371 



SECTION LV. 

THE WATER VOLTAMETER. 

Apparatus required : Water Voltameter, two storage cells, 
tangent galvanometer (with a constant of \ or 2), reversing key 
and resistance coils. 

When an electric current decomposes a liquid through which 
it is sent, the liquid is called an " electrolyte," and the process 
of decomposition is called " electrolysis." The laws of electro- 
lysis state that the weight of an element set free from a com- 
pound by electrolysis, is proportional to the current, the time, 
the atomic weight and inversely proportional to the valency of 
the element. One ampere flowing for one second liberates, e.g. 
•01118 grams of silver from any silver salt. The amount of an 
element liberated by a current in a given time may therefore be 
used to determine the magnitude of the current and hence to 
standardise a current-measuring instrument. 

In the present exercise a tangent galvanometer is to be 
standardised by means of a water voltameter. There are two 
ways of using a water voltameter, according as the hydrogen 
alone is measured, or both oxygen and hydrogen are collected 
together. In the latter case we gain by having a greater volume 
of gas to measure and being able therefore to reduce the time 
of the experiment, but there is some danger of error owing to 
the formation of ozone and its absorption in the water. It has 
however been shown by Kohlrausch that if currents of over one 
ampere are to be measured with an accuracy sufficient for 
commercial purposes, a voltameter, in which both gases are 
collected together, will answer the purpose. For smaller 
currents the hydrogen only should be measured. The difficulty 
may also be overcome by using a solution of sodium hydrate 
(between 5 and 10 "/o) with nickel electrodes. 

The volume of mixed gas produced by the passage of an 
ampere for a second is 1734 c.c. at normal temperature and 



LV 



THE WATER VOLTAMETER 



263 



pressure. The gas is collected in a graduated tube and its 
volume measured. To reduce the observed volume to the 
normal conditions, the temperature of the gas must be measured 
by a thermometer placed, if possible, inside the voltameter tube. 
The pressure is obtained by reading the barometer, by applying 
if necessary a correction for a, difference of pressure inside and 
outside the voltameter tube, and by deducting the pressure of 
aqueous vapour in the tube. As pure water offers a high resistance 
to the electric current a little sulphuric acid is added to it. If 
the reductions are to be made to the highest possible accuracy, 
account must be taken of the fact that the pressure of saturated 
vapour of water is less over a solution of sulphuric acid than 
over pure water. But in the present exercise it is assumed 
that the acid added to the water is not sufficient to produce a 
sensible effect. 

The voltameter provided (Fig. 105) consists of two portions, 
a lower reservoir and the graduated tube, which fits into it by 




Fig. 105. 

means of a giound joint. The lower reservoir communicates 
with the atmosphere through an opening which can be closed 
by a glass stopper. During electrolysis this stopper must be 
removed. Pour water acidulated with about 15% of dilute 
sulphuric acid into the reservoir so as to render it about three- 
quarters full. Insert the tube and stopper and tilt the volta- 
meter so that the liquid runs into and completely fills the tube. 
Place the instrument in a tray and remove the stopper. 

Connect the galvanometer and voltameter in series with a 
reversing key, a resistance and two storage cells. The object of 



264 MAGNETISM AND ELECTRICITY hV 

the reversing key is to correct for errors of zero of the tangent 
galvanometer, by taking its deflections during half the time in 
one direction and half the time in the other. The key might 
therefore be inserted so as to affect the galvanometer only, leav- 
ing the current through the voltameter in the same direction 
throughout the experiment. 

Adjust the resistance so that when the circuit is closed, the 
deflection of the galvanometer needle is about 45". 

When this has been done, interrupt the current at the key, 
and wait till the bubbles of gas in the upper part of the column 
of liquid have ascended into the space above. Then read the 
position of the top of the column on the tube, measure to the 
nearest cm. its height above the liquid in the base, and note the 
temperature of the gas. Read the zero of the galvanometer. 

Now make the circuit at the key, noting the time, and at the 
end of a minute read the galvanometer, repeating the reading 
every two minutes till the tube is about J full of gas. One 
minute after taking the last reading, break the circuit, noting 
the time. After allowing the bubbles of gas in the column of 
liquid to ascend, read the position of the top of the column 
and measure as before the height of the liquid column, and 
the temperature of the gas. 

At a given instant make the circuit again, so that the current 
passes in the opposite direction through the galvanometer for 
an equal time. Read the galvanometer every two minutes as 
before, and the final position of the level of the liquid. 

Read the barometer. 

Calculate for each of the three observations of the column 
of liquid, the volume of gas reduced to normal temperature and 
pressure, and subtract the first from the second and the second 
from the third. The diflerences are the volumes of gas produced 
in the observed times by the passage of the current. 

Take the mean of the tangents of the galvanometer deflec- 
tions for each period. Then if d be the mean tangent, and K 
the constant of the galvanometer, the average current passing 
is Kd amperes. If v c.c. of gas were liberated in t seconds, 
then v=V. Kd . t, where V is the volume of mixed gas liberated 
by 1 ampere in 1 sec, both being measured at the same tempera- 
ture and pressure. 



LV 



THE WATER VOLTAMETER 



265 



If V and V are the volumes reduced to a pressure of 76 cm. 
and a temperature of CC, we may substitute for Fits numerical 
value 1734 and thus find 

V 



K = 



Calculate K from the observation in this way, arranging the 
work as shewn below. 

The two sets of observations are reduced separately so as to 
afford a check on the calculations and a test of the consistency 
of the results. 

Remove the tangent galvanometer, and substitute for it one 
of the boxes used in the Magnetic Survey of the Laboratory 
(Section L.). Determine the time of oscillation of the needle. 
Determine also the time of oscillation of the needle when placed 
at some point of the Laboratory at which the earth's horizontal 
magnetic force H is known, and from the observations calculate 
the value of H at the place where the galvanometer stood. 

From the value of H and the radius and number of turns of 
the coil calculate the constant of the galvanometer (see Section 
LL), and compare the calculated value with the result of the 
experiment. 

Arrange your observations as follows : — 

17 March, 1898. 

Voltameter A. Tangent Galvanometer B. 

Height of Barometer = 75*67 cms. 



Voltameter observations 


I 


n 

133-5 


m 


1 Heading of voltameter tube*, c.c. 


16-5 


253-8 


Heiglit of column of water, cms. 


35 


26 


15 


„ „ equivalent mercury column 


2-58 


1-91 


111 


Pressure of ga.s, cms. 


7309 


73-76 


74-56 


Temperature (Centigrade) 


18° -8 


18''-7 


18''-5 


Corresponding va|K)ur presHUi*e, cms. 


1-61 


1-60 


1-68 


Presssure of dry gas, cms 


71-48 


7216 


72-98 


Volume of gas reduced to 76 cms. 








and O'C, ac 


14-5 


118-6 


228-7 



* If the tube is not divided into oubio oentimetret, the value of the divinont 
onght to be ascertained by experiment. 



266 



MAGNETISM AND ELECTRICITY 

Galvanometer Observations I. 



LV 



The readings of the ends of the pointer, marked A and B, 
are taken as positive when the deflections are counter-clockwise 
as measured from the zero divisions of the scale. 



Time 


Readings of Pointer 


Deflections 


Tangents 


End A 


End B 


Mean 


11 h. 57 m. 


+ o°-o 


+ 0°-2 


H-O^-l 






11 58 

12 00 
02 
04 
06 
08 


+ 46-6 
46-2 
45-5 
45-5 
45-3 
44-9 


+ 46-6 
46-3 
45-7 
45-7 
45-5 
451 


+ 46-6 
46-2 
45-6 
45-6 
45-4 
45-0 


46-5 
46-1 
45-5 
45-5 
45-3 
44-9 


1-054 
1-041 
1-018 
1-018 
1-010 
0-996 


09 


+ 0-0 


+ 0-2 


+ 0-1 














Mean = 


1-023 



Current started at 11 h. 57 m. 
„ stopped at 11 h. 08 m. 

Similarly for the observations during the second half of the 
experiment. 

The results are tabulated as follows : — 



Volume generated 


Time 


Mean 
tangent 


K 


118-6- 14-5 = 104-1 
228-7-118-6 = 110-1 


660 sees. 
650 „ 


1023 
1-050 


-89 
-93 






Mean = 


•91 



Constant of Galvanometer by calculation 

Time of oscillation at place of observation ... = 7*12 sees. 

„ „ „ standard position .. . ... = 7-16 „ 

Value of jy „ „ „ = -171 „ 

Value of ^ at place of observation ... ... = '173 „ 

Radius of Coil of Galvanometer ... ... =9*7 cms. 

Number of turns ... ... ... ... ... = 3 

.'. Constant = ... ... ... ... = "89 



SECTION LVI. 

STANDARDISING CURRENT METERS BY THE 
COPPER VOLTAMETER. 

Apparatus required : Current-measuring instrument to he 
standardised (an instrument intended to measure accurately one 
or two amperes is suitable) ; storage cells, copper depositing cell, 
copper electrodes, plug-key, resistance coils, accurate balance. 

In the previous exercise the amount of electrolysis produced 
by the passage of a current was estimated by the volume of gas 
generated. More accurate results are obtained by the use of an 
electrolyte of which one of the products of decomposition is a 
solid which can be weighed. Silver and copper salts are found 
to be suitable for the purpose, as secondary reactions may be 
more completely avoided than in the case of other metals. For 
the most accurate work the silver voltameter is used*, but its 
manipulation requiring great care, and the materials being ex- 
pensive, copper may be substituted, and with proper precautions 
an accuracy of one part in a thousand may be attained. 

In the present exercise a solution of copper sulphate in 
water will be used, and a current meter will be standardised 
by the weight of copper deposited. The example is worked 
out on the supposition that the instrument to be standardised 
ifl a Kelvin Current Meter, but any other accurate instrument 
such as that shewn in Fig. 106 may be substituted. 

Prepare a 20% solution of copper sulphate, by adding 120 
grams of copper sulphate crystals to 480 grams of water. When 
the crystals have dissolved, filter into the beaker provided and 
add one or two c.c. of strong sulphuric acid. 

• Lord Rayleigh and Mm Sidgwick ♦•On the electrochemical equivalent of 
silver and on the absolute electromotive force of Clark cells," Phil. Tram. 1884. 



268 



MAGNETISM AND ELECTRICITY 



LVI 



Clean the three copper plates provided, two of which are to 
serve as anodes and the third as cathode, with sandpaper. Dip 
the one which is to serve as cathode into dilute nitric acid for 
about three minutes, then remove it and place the three in 
dilute sulphuric acid for about three minutes, finally wash 
under a stream of tap-water. 

Place the first two plates in the side clips A of the stand 
(Fig. 106) provided, and the third plate in the centre clip. 




Fig. 106. 

Join up the copper voltameter, a plug-key K and an adjust- 
able resistance to the instrument to be standardised, through 
two storage cells (one of which only is shewn in the figure) or 
other battery of small resistance giving about 4 volts. Be 
careful to connect the terminals of the battery in such a way, 
that the current passes from the outside plates (anodes) to the 
central plate (cathode) of the voltameter. If the instrument to 
be standardised gives deflections in different directions when the 
current is reversed, a reversing key should be inserted so that 
the directions of the current can be changed rapidly in the 
current meter without being reversed in the voltameter ; and that 
change should be made in the middle of the experiment. 

Adjust the resistances until the current is of convenient 
amount to be measured. 

Raise the plates out of the solution, take the copper cathode 
from its clip, wash it in clean water, and dry first in a sheet of 
filter-paper, then before a fire, heating coil, or gas flame, taking 
care not to heat the plate appreciably. When the plate is dry 
and at the temperature of the room weigh it carefully. 



LVI COPPER VOLTAMETER 269 

As the absolute weight is required, the plate must be 
weighed on both sides of the balance according to the method 
describetl on p. 45, and trustworthy weights only must be used. 

Read the zero of the instrument. 

Replace the cathode and lower the copper plates into the 
solution, make the circuit at a time to be noted in minutes and 
seconds, and as soon as possible take a reading of the instrument 
to be standardised. 

Take readings every five minutes for forty-five minutes or 
an hour. 

The true readings will be the differences between the actual 
readings and the mean of the zeros read at the beginning and 
end of the experiment. 

Note the temperature of the electrolyte. 

While the electrolysis is going on, clean a beaker, fill it with 
clean water, and add a few drops of sulphuric acid. 

At the end of the above period break the circuit, noting the 
time accumtely, and read the zero of the instrument. 

Raise the electrodes, remove the cathode and dip it as 
quickly as possible into the acidulated water, then hold it 
under a gentle stream of water from the tap for a minute. 
Now dry it as before, first in a pad of clean filter-paper, then 
before a fire, heating coil or flame, and after cooling weigh again 
accurately. 

The relation between the current (7, the electrochemical 
equivalent of copper z^ and the weight W deposited in a given 
time t is 

W 
W--C.z.t, .-.(7= ,. 

z . t 

In the absence of all secondary reactions the value of z 
should be constant. But owing probably to the presence of 
dissolved oxygen, the amount of copper deposited is not strictly 
proportional to the current and depends to a small extent on 
the temperature. These effecta can be taken into account by 
making z depend on the current density at the cathode and on 
the temperature. Thomas Gray, who has carefully investigated 
the amounts of copper deposited under different conditions, has 



270 



MAGNETISM AND ELECTRICITY 



LVI 



given the following values for the electrochemical equivalent 
of copper. 



Area of cathode in 






square centimetres 
per ampJsre of 


Temperature 12° 


Temperature 23° 


current 






50 


•0003288 


•0003286 


100 


•0003285 


•0003283 


150 


•0003282 


•0003280 


200 


•0003279 


•0003277 



The results of this Table may be expressed by the equation 

A KQ f TO 

•0003288 - 0000003 ,^ - 0000002 --— , 
50 11 

where A is the area of the cathode surface per ampere, and t 
the temperature. The value of A is found by calculating the 
current in the first instance approximately, using ^-^ as 
the equivalent. 

Measure the total area of the two sides of the cathode, and 
obtain an approximate value of the area per ampere. From 
this and the temperature during the experiment find the equi- 
valent which applies to your experiment. 

B,ecord and reduce the observations as follows : — 



15 June, 1899. 






Weight of cathode before electrolysis 


.. = 


20-3798 grams 


„ ,, ,, atter ,, 


.. = 


21-5786 „ 


Amount of copper deposited 


. = 


1-1988 „ 


Duration of experiment =lh. Om. 25 s. . 


= 


3625 seconds 


^ ^ . ^ , 1-2 X 3000 
Current approximately -q^qq- 


. = 


1 ampere 


Total area of cathode 


= 


66 sq. cm. 


Area of cathode per ampere of current . 


. = 


66 


Temperature 


. = 


18° C. 


Equivalent of copper 


.. = 


•0003286 


^ 1-1988 






Current =.^^32gg^ 3^25 - ' 


• = 


r0064 ampere 



LVI 



COPPER VOLTAMETER 



271 



Time 


Beading 

of 
Kelvin 
balance 


Zero 


True 
reading 


Square 
root 


llh. Om. current made 
5 

10 
15 

50 
55 
12h. Om. 25 8. current broken 


409-6 
406-6 
406-4 
406-4 

405-4 

405-6 
406-4 


•2 
•2 


409-4 
406-4 
406-2 
405-2 

405-2 
406-4 
406-2 


20-23 
20-16 
20-15 
20-13 

20-13 
20-16 
20-15 








MeaD = 


2014 



Constant of balance = 



Mean current 



10064 



Mean square root of deflection 20*14 

= 04997 
Constant given by instrument maker ... ... = 05000 

Thus currents measured by the instrument using the con- 
stant -05 would agree to within one part in a thousand with 
their value as determined by copper electrolysis. The constant 
supplied by the maker of the Kelvin balance is equal to the 
mean current divided by twice the square root of the reading ; 
i.e. '020 in the case of the above instniment. A table of doubled 
squared roots is provided with the instrument, and the last 
column in the above Table may be replaced by one in which 
the double square roots are entered. But the method here 
adopted is applicable to all instruments of the dynamometer 
type and the square roots are easily found in Barlow's Tables*. 
If the instrument to be tested is a direct reading one, the last 
column is unnecessary, and if it is of the tangent galvanometer 
type it must be replaced by the tangents of the angles of 
deflection. 



* If the square root of e.g. 408-4 is taken from Barlow's tables, interpolation 
would seem necessaiy between the values of the roots given for 408 and 409, but 
this may be avoided by finding that number in the neighbourhood of 2000, the 
square of which has for its first four significant figures 4084. The ponition of 
the decimal point is obvious. 



SECTION LVII. 

ON MIRROR GALVANOMETERS AND THEIR 
ADJUSTMENT. 

In Section LI. it has been explained that the sensitiveness 
of a tangent galvanometer may be increased by diminishing the 
radius of the coil through which the current is passing; but 
that if the tangent law is to hold good the length of the needle 
must also be diminished. In many experiments it is of greater 
importance to have the galvanometer as sensitive as possible, 
than that the tangent law should hold accurately. We use in 
such cases a galvanometer, the coils of which have as small a 
radius as possible, leaving only just sufficient room for the 
magnet to move inside. The movement of this magnet is 
rendered apparent, and measured if necessary, by the help of 
a mirror attached to it, in the way explained in Section XXXIY. 
Several small magnets are often used instead of a single oue in 
order to increase the magnetic moment, and these are either 
attached directly to the back of the mirror by means of a little 
shellac, or the mirror is placed outside the coil and attached to 
a thin wire which passes through the coils and carries the 
magnets at its end. The suspended system must be free to 
move round a vertical axis, and is for this purpose attached to a 
fibre just strong enough to carry its weight and as free from 
torsion as possible. 

The deflection of a galvanometer for a given current being 
increased by a diminution of the intensity of the magnetic 
field at the centre of the coil we have a further means at our 
disposal of rendering a galvanometer more sensitive, or of ad- 
justing its sensitiveness to any desired value within certain limits. 
For this purpose a permanent magnet is used which together 



LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 273 

with the earth's field produces a resultant field of the desired 
strength. A similar magnet is also necessary, when the galva- 
nometer has to be set up in such a position that the plane of 
the coil does not lie in the magnetic meridian. The strength 
and direction of the earth's field being given, it is theoretically 
always possible to produce a resultant field of any desired 
strength and direction by placing a permanent magnet with a 
sufficiently great magnetic moment in a proper position. When 
gi-eat sensitiveness is required, and at the same time the 
direction of the field is to be altered, it is generally advisable to 
proceed by two steps. With a magnet of known moment, it 
will not be difficult to find by calculation a position for it 
such that it will nearly neutralise the earth s field. A second 
weaker magnet may then be used to regulate the strength 
and direction of the field. A galvanometer made sensitive by 
counterbalancing in this way the greater part of the earth's 
force by means of external magnets has the disadvantage of 
being very sensitive to slight magnetic disturbances either 
caused by actual changes of the terrestrial forces, or to 
disturbing currents (electric trams) or to the accidental dis- 
placements of magnetic material (keys, spectacles, corset steels) 
which are almost unavoidable in a laboratory. To get rid of 
that portion of these effects which is nearly uniform in the 
space occupied by the galvanometer needles, so called astatic 
magnetic systems are often used. These consist of two mag- 
nets or two sets of magnets, which have magnetic moments 
of nearly equal value, and are rigidly connected with each other, 
so that the similar poles of the two sets point in opposite 
directions. One of the sets is placed in the centre of the 
galvanometer coil, the other above or below it, and in some 
instruments the second set will be placed in the centre of a 
second galvanometer coil in all respects similar to the first. 
The whole of the combination of magnets, called an astatic 
system, will behave like a very weak magnet towards the 
earth's force, and may set in a direction which it is impossible 
to predict. 

To make this latter point clear, let H represent the earth's 
field, m^.m^ the magnetic moments and a,, Og the angles between 
8. p. 18 



274 MAGNETISM AND ELECTRICITY LVII 

the magnetic meridian and the magnetic axes of the two sets. 
The mechanical couples exerted by the earth's field will be 

Hm^ sin a-y and Hnu sin a.^ 

and in the position of equilibrium 

mi sin «! + W2 sin ao = 0. 

If the angle between the two magnetic axes is very nearly 
equal to two right angles we may write 

^2 = «! + TT + 5 

where 8 is a small angle, so that 

sin as = — (sin oti -t- 8 cos a^). 
Hence for equilibrium 



lai 




h 




7?li 


- 


ma 


nil 


= mo 






«! 


IT 

~ 2 ' 







If 

it follows that 

hence the astatic system will set at right angles to the magnetic 
meridian) if on the other hand S = but m^ differs from mj, 
oti = and the system will set in the meridian. As in practice 
neither h nor m, - m^ vanish, but will be small, «] depends on 
the ratio between two unknown small quantities. Directing 
magnets will have to be used with an astatic system, in order 
to bring it into its proper position with respect to the galvano- 
meter coils and to regulate the strength of the field. 

The increased sensitiveness of an astatic galvanometer de- 
pends on. the fact that while the two sets of magnets oppose 
each other in so far as the earth's directing force is concerned, 
the electric current acts on both in the same direction. If 
there are two galvanometer coils, the current is led through 
them in opposite directions, so that the couples exerted by 
the currents in both coils have the same direction. A galvano- 
meter with two coils can also be used as a "differential galvano- 
meter," when it is required to test the equality of two currents 
or to measure very small differences between them. The two 



LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 275 

cun-ents are in that case sent separately through the two coils, 
so that their eflfects on the suspended system of magnets oppose 
each other. 

While the observer has it in his power to increase consider- 
ably the sensitiveness of a galvanometer by weakening the 
magnetic field, he must remember that in so doing, he will 
increase the peiiodic time, i.e. the time of oscillation of the 
needle, and thereby render its motion more sluggish. This 
to a great extent counteracts the advantage gained by the 
increased sensitiveness, especially when owing to disturbing 
causes the zero of the needle is a little unsteady, for it becomes 
in that case impossible to distinguish a true deflection from an 
accidental shift. Therefore in modem galvanometers intended 
to be highly sensitive, great importance is attached to having 
the suspended magnetic system as light as possible, a small 
moment of inertia allowing a corresponding weakening of the 
directing field without increase of the time of oscillation. 

In the case of a D'Arsonval galvanometer or any instrument 
with a suspended coil, it is not generally in the power of the 
observer to alter the sensitiveness. 

Students should pay careful attention to the optical arrange- 
ment on which the measurement of the angles depends. If the 
objective method is used, they should secure a good image of 
a wire on the scale, and in every way try to increase the ac- 
curacy of reading. It must be remembered that doubling the 
accuracy of reading doubles the effective sensitiveness of the 
galvanometer just as much as halving the strength of the 
directing field, without introducing the disadvantage of increased 
peri<xlic time. Insufficient care is often given by instrument 
makers to the mirror of a galvanometer, which is a more im- 
portant part of the instniment than many others, to which 
they devote great attention. The image formed should be as 
perfect as the size of the miiTor will allow; the latter should 
be either plane or concave with a radius of curvature of 
about one metre. If the instrument is home made, microscope 
cover glasses are generally silvered for the purpose, and among 
a number of them, some are always found which give sufficiently 
good images before mounting, but after the magnets have been 

18—2 



276 MAGNETISM AND ELECTRICITY LVII 

fixed to the back of the mirror, it often turns out to be useless 
on account of the distortion produced by the material which is 
used to attach the magnets. It is therefore better to fix these 
in the first place to thin mica, and to attach the mirror to the 
other side of the mica by means of a small drop of shellac 
holding the mirror, at one point only. If the mirror hangs out- 
side the galvanometer coil the difficulty does not arise. The 
curvature of the mirror, and the definition of the image formed 
by it, may be investigated by using a source of light some 
distance away, and determining whether there is a good re- 
flected image within a few feet of the mirror. If there is, 
the distances of object and image will determine the radius 
of curvature. If the mirror is plane, or nearly so, the image 
may not be real or may be too far away, and in that case, must 
be looked at through a telescope. The telescope being focussed 
for the image, we may easily determine the distance of a point 
which when looked at directly is also in focus. From this and 
the distance of the source of light, the focal length of the mirror 
may be determined. 

The principal adjustments of mirror galvanometers consist 
according to the above explanations, (1) in securing that the 
suspended system can turn freely round a vertical axis through 
the largest angles on either side of the zero position, which 
are likely to be used during the observations, (2) in securing 
that the resultant force should be nearly parallel to the plane of 
the galvanometer coils, and (3) in adjusting the strength of the 
field to the required intensity. 

If it is desired to adjust a galvanometer ah initio, the first 
step should be to place the galvanometer on a firm support, if 
possible in the centre of the room so that access may be had 
to it from all sides. Freedom of motion of the galvanometer 
needle should be obtained by adjusting the level of the in- 
strument. If it cannot be secured thus, carefully notice what 
is the cause of the impediment. Possibly the needle hangs 
too low or too high, and in that case the suspension must be 
altered, but this should only be done by experienced hands. 
When the needle is free, observe its approximate position of 
rest. With non-astatic systems it should be in the direction 



LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 277 

of the resultant field at the place of observation. If this is not 
the case, the fibre is probably twisted. Unless the angle of 
twist is very great, no serious error will result ; but instruments 
should be provided with some means of turning the suspension 
head, and thus untwisting the fibre, so that the needle may 
hang in the proper direction. It is always an advantage to 
be acquainted with the peculiarities of each instrument, and 
the effects of torsion should be noted by twisting the upper 
end of the suspension through a measured angle and noting the 
angle through which the suspended magnet turns in conse- 
quence. This angle of twist will prove useful in the future use 
of the instrument, because if the suspended system is losing 
its magnetisation, the angle will gmdually increase, and shew 
when the time has arrived for taking out the suspension and 
re magnetising the magnets. 

The time of oscillation T of the magnets when swinging in 
the earth's field should also be noted. If M is the magnetic 
moment, H the strength of the field, / the moment of inertia 
and T the torsion per unit angle of twist we have the relation 



y HM- 



and if a twist through an angle a at the upper end of the suspen- 
sion has produced an angular displacement 6 of the magnet 
system : 

T{a-e) = HM^md, 

where o will in general be sufficiently great compared to 6 to 
allow the latter to be neglected on the left-hand side of the 
equation. If the suspension has been put together in the labo- 
ratory, the weights and dimensions of different portions of it 
should have been noted, so that the moment of inertia can be 
calculated with sufficient accuracy. In that case, the above 
relations will allow the determination of J/ and t. 

The value of t is of some interest in instruments intended 
for delicate work, for it determines the limit beyond which it is 
not possible to push the sensitiveness of the galvanometer, even 
if the earth's field be wholly neutralised. 



278 MAGNETISM AND ELECTRICITY LVII 

The time of oscillation and the direction of the needle having 
been noted, the position where it is desired to set up the galva- 
nometer must be considered. This will generally be along a 
wall of the laboratory, or at right angles to it, and therefore not 
necessarily in the direction of the magnetic meridian. A rough 
calculation will give the strength of the required magnetic field 
which together with that of the earth will set the magnet 
parallel to the galvanometer coils, and give a time of oscillation 
which should not be greater than 8 or 10 seconds. If the focal 
length of the mirror is about a metre, the scale may be set up 
at that distance from the mirror, but if the mirror is plane a 
convex lens of one metre focus should be fixed to the galvano- 
meter as near to it as possible. If the mirror is neither plane 
nor has the required radius of curvature, the focal length of a 
lens must be calculated so that, when fixed near the mirror, a 
point of light at a distance of one metre should give a real 
image at the same distance. The scale is now set at the proper 
distance, and the last adjustment of focus is made either by 
altering slightly this distance or by introducing a weak lens to 
alter the divergence of the beam incident on the mirror. A 
thin wire placed in the incident beam, which may also be made 
movable for greater facility of focussing, serves as the object, 
the image of which on the scale determines the angular position 
of the mirror. 

Very often the preceding adjustments have been made for 
the student, at any rate approximately, and in that case his first 
care should be to check the adjustment for focus and to improve 
it when necessary. If the instrument is to be used not simply 
as an indicator but to measure deflections, the position of the 
scale must also be looked to. It should be placed at right 
angles to the line joining its centre to that of the mirror. This 
is most easily done, as explained in Section XXXIV. by measur- 
ing the distance from the ends of the scale to the mirror, or if 
the galvanometer is covered by a glass shade, it will be sufficient 
to measure the distances from that point of the shade, which is 
estimated to lie on the central line. The investigation on page 
159 shews that if an accuracy of one per cent, in the readings 
is aimed at, and the means of deflections to right and left are 



LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 279 

always taken, the scale may be inclined as much as 8 Vo without 
doing any harm. The difference in the distances of the ends 
of the scales from the mirror should in that case not exceed the 
seventh part of the length of the scale. If however the deflec- 
tions to either side should be correct to within one per cent, a 
ten times greater accuracy is required in the adjustment of the 
scale and this requires some care in the measurements. 

The last adjustments relate to the sensitiveness of the 
instrument. If this is insufficient a permanent magnet should 
be placed parallel to the galvanometer needle and so that the 
line joining the centres of the two magnets is approximately 
perpendicular to their magnetic axes. If the poles of the out- 
side magnet point in the same direction as those of the sus- 
pended needle, the magnetic field will be weakened by it. The 
magnet is first placed at a distance and then brought slowly 
nearer, the spot of light being watched at the same time, so as 
to keep it near the centre of the scale. It will be found that 
when the outside magnet is brought too near, the position of 
the galvanometer needle becomes unstable and tends to be 
reversed. When the magnet is placed so far away that the 
needle is just stable, the instrument is as sensitive as it can be 
made, but it will probably not be advantageous to use it in 
this most sensitive condition owing to the increase in the length 
of the time of oscillation. 

In some instruments the controlling magnet is clamped 
to a vertical rod fixed to the galvanometer stand, and its position 
is altered by loosening the clamp and sliding the magnet up- 
wards or downwards. Some makei-s have adopted a method in 
which two controlling magnets at a fixed distance are used. 
The field due to the two magnets may be strengthened or 
weakened by altering the angle between their magnetic axes. 
In the case of galvanometers having astatic systems, the sensi- 
tiveness may be altered in the same manner by an outside 
magnet. 

Delicate galvanometers may be damaged when too strong a 
current is sent through them ; and very often it is not possible 
at the beginning of an experiment to make sure that the elec- 
tromotive force at the terminals of the galvanometer does not 



280 MAGNETISM AND ELECTRICITY LVII 

exceed the limit of safety. In that case a "shunt" should be 
used. This is a resistance smaller than that of the galvanometer 
and connected to it in such a way that the current will pass 
through the galvanometer and shunt in parallel. Sometimes a 
number of shunts are supplied with the instrument having their 
resistances graduated so that only the j'^, yj^ or jj^jj^th. part of 
the current may pass through the galvanometer. 

The sensitiveness of a galvanometer is measured by its 
power to indicate small variations of current, but it does not 
follow that the most sensitive galvanometer is the one that 
indicates the smallest variations of the quantity to be measured. 
On the contrary a galvanometer, which in the ordinary meaning 
of the word would not be called " sensitive " at all, is sometimes 
the one which will give the most accurate results. As it is 
very important that the student should clearly realise this, it is 
necessary to enter a little further into the theory of the galvano- 
meter. Supposing the annular space, which is to contain the 
galvanometer coil is given, we may fill it either with thin or 
with thick wire ; in the former case we shall get a " sensitive " 
galvanometer, but one having high resistance, while the thick 
wire will give a low resistance but smaller sensitiveness. It is 
on the relation between the resistance and the sensitiveness 
that the theory of the galvanometer depends. If starting with 
a thick wire, we wish to replace it by one of half the diameter, 
we should be able to replace each turn by four of the thinner 
wire. But four turns of wire having one quarter the cross 
section will have sixteen times the original resistance. Hence 
while we have increased the sensitiveness four times we have 
increased the resistance sixteen times. In a similar way it may 
be proved quite generally that — neglecting the space lost by 
the insulation of the wire — the sensitiveness of galvanometers 
having the identical spaces filled by windings, will vary as the 
square root of the resistance. 

Writing k^/G for the sensitiveness of a galvanometer where 
G is its resistance and k a constant depending on the shape 
and size of the annular space containing the windings, let a 
circuit containing an electromotive force E be formed by the 
galvanometer and a resistance R, the current will be JE/(Ii + G) 



LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 281 

aud the deflection of the galvanometer fcE*JG/{R + G). The 
deflection of the galvanometer will therefore be very small 
when the resistance of the galvanometer is very small and also 
when it is very large, and it may easily be shewn that the 
deflection will be a maximum, when G = R. Hence when small 
electromotive forces have to be measured, that galvanometer 
will be most delicate which has the same resistance as the 
external resistance. For thermoelectric work with circuits which 
generally have very small resistances, so called "sensitive" 
galvanometers will not be sensitive at all, and low resistance 
galvanometers will give the best results. When the cii'cuits 
are more complicated as in the Wheatstone bridge arrangement 
the above investigations have to be extended, if it is desired to 
find the most suitable galvanometer, but the general rule will 
be found to hold, that when the external resistances are small, 
the galvanometer resistance should be small also. 

In the case of galvanometers, in which the poles of the sus- 
pended magnets come close up to the windings, it cannot be 
assumed that the currents are proportional to the angles of de- 
flections, even when these are small. If therefore small currents 
passing through the instrument are to be compared with each 
other, the instrument should be calibrated. This is most readily 
done by the method used in Section LIII. (Exercise I.), a current 
being sent through a uniform wire stretched along a graduated 
scale and a suitable resistance. If the terminals of the galva- 
nometer are brought into contact with the wire, the resistance 
of which should be small compared to that of the galvanometer, 
the currents passing through that instrument will be propor- 
tional to the length of wire included between the terminals. 

A short description may be given in conclusion of the 
"damping" of galvanometers. A current passing through the 
instrument will deflect the needle, which will begin to oscillate 
about its new position of equilibrium. The oscillations follow 
the same laws as those of a pendulum, and will gmdually di- 
minish owing to frictional resistance and electrical damping. 

In many cases it is desirable that the needle should come to 
rest quickly, for which purpose it is advisable to increase the 
resistance, either mechanically by attaching a vane to the 



282 MAGNETISM AND ELECTRICITY LVII 

suspended system, or electrically by bringing a mass of well- 
conducting material (copper) near the oscillating magnets. The 
currents induced in the copper by the motion of the magnets 
react on the latter and oppose its motion (Lenz' law). 

When the damping is so great that the needle will not 
oscillate at all, but gradually takes up its new position of equili- 
brium, without passing beyond it, the motion of the needle is 
called "aperiodic," and the galvanometer is said to be "dead 
beat." 

When the instrument is used to measure the quantity of 
electricity conveyed by a large current of short duration (dis- 
charge of a condenser, induction kicks), we require the deflection 
which would be produced if there were no friction, and in that 
case the damping should be small, and must be taken into 
account (see Section LXIL). A galvanometer used in this 
fashion is called a ballistic galvanometer. The angular dis- 
placement of a dead beat galvanometer is represented b}^ the 
expression 

c(l-e-^% 

where c is the ultimate displacement, which as is seen, will be 
reached theoretically only after an infinite time. 

The angular displacement of a damped needle, measured 
from its position of equilibrium, is represented by the equation 

00 = ae--^^''^ cos K{t — U), 

T being the time of oscillation, and t^ any time at which the 
needle has its greatest positive elongation. 

In case the numerical value of \ has to be determined, suc- 
cessive turning points have to be observed. 

If tx is the time of the first observed positive elongation {x^ 
the successive positive elongations will take place at times 
ti + T, ti-^ 2T etc. and will therefore be given by 

a?3 = ae-'-'^(^+2r,/r 



LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 283 

Taking logarithms on both sides and then diiferences between 
successive equations we obtain 

log Xi — log x» = 2\ log e 

log Xi - log a^s = 2\ log e 

log .Tj — log x^ = 2\ log e. 
Hence the differences between the logarithms of successive 
elongations are constant and the quantity X is called the " loga- 
rithmic decrement." The value of X determined in this way 
may be affected by a considerable error if the zero reading of 
the galvanometer changes progressively. This eiTor may be 
eliminated by measuring the elongations on both sides and 
taking the mean of the values of \ as found from the positive 
and negative elongations. A manner of reducing the obser- 
vation, which is better, because it saves some arithmetical 
labour, consists in measuring successive arcs of swing from 
right to left, left to right and so on. Calling these successive 
arcs: s^, 5«, etc., we find 

log s, - log Sn = (n-l)X log e 
log 52 - log Sn+i = (« - 1) \ log e, 
so that a series x>f independent values of X may be obtained the 
mean of which is taken for the final result. When the oscilla- 

s ^ s 
tions diminish slowly so that the square of ^— " may be 

neglected, the first of the above equations simplifies to 

~(n-l)Sn' 

The quantity \ occurs chiefly in physical measurements, when 
the first deflection of a galvanometer needle is observed and it 
is desired to calculate what that deflection would have been in 
the absence of all damping. As the time taken by the needle 
to pass from the position of rest to its first elongation is T/4(, 
the above equations give a = Xie~^^^, where x^ is the observed 
elongation and a the required amplitude. If X be small this 
reduces to 

an equation which we shall have occasion to use. 



284 MAGNETISM AND ELECTRICITY LVII 

Exercise I. Set up the mirror galvanometer in the given 
position and measure the time of oscillation of the suspended 
magnet, when oscillating under the action of the earth's force 
only. Double the time of oscillation by means of an external 
magnet. Determine the electromotive force necessary to pro- 
duce an angular deflection of 'Ol. Hence, assuming the tangent 
law to hold, calculate, using the given value of the galvano- 
meter resistance, the magnetic field produced at the poles of 
the suspended system per unit current passing through the 
galvanometer coil. 

Exercise II. Calibrate the galvanometer indications, and 
prepare a table shewing the relation between the deflection of 
the mirror and the current passing through the instrument. 



SECTION LVIIL 

MEASUREMENT OF RESISTANCE. . 

THE POST OFFICE FORM OF WHEATSTONE'S 
BRIDGE. 

Apparatus required : Galvanometer and scale, Post Office 
resistance boa-, Leclanche cell, coils, voltmeters, platinoid mire, 
and connecting luires. 

The students are supposed to be familiar with the principle 
of a Wheatstoue Bridge, and to have had some practice in the 
measurement of resistance by the simpler forms of bridge 
arrangement. When two of the resistances of the bridge are 
formed by a stretched wire, with a movable sliding contact for 
the junction leading to the battery or galvanometer, incon- 
veniences arise owing, (1) to the fact that the sum of the two 
resistances is fixed, being the total resistance of the wire, while 
increased accuracy could often be obtained, if that resistance 
could be varied according to the value of the resistance to be 
measured ; (2) to the uncertainty of the measurements, due 
to the possibility of different parts of the wire having unequal 
resistances, or the contacts being faulty; (3) to the arith- 
metical labour involved in reducing the measurements, which 
is of no importance if the instrument is only used occasionally, 
but becomes irksome by accumulation if many measurements 
are taken ; (4) the want of portability of the instrument. The 
requirements of telegraph offices, which have been the cause of 
many improvements in electrical instruments, have led to the 
construction of a form of a bridge consisting of a series of 



286 



ELECTRICITY AND MAGNETISxM 



LVIII 



resistances, appropriately connected, and compactly placed to- 
gether in a box which is known as the " Post Office box " 
(Fig. 107 a and b). 




Fig. 107 (I. 



Fig. 107 b. 



To understand its construction students are referred to 
the diao^ram of the bridge (Fig. 108) in which the battery 
and galvanometer circuits are brought 
to spring contacts at K^, Kc,. These 
spring contacts are placed in the box 
as shewn in the figures. There is a 
metallic connexion inside the box, not 
shewn in the figure, between C and Ko 
and also between A and K^. The letters 
in Figs. 107 6 and 108 correspond, so that 
if the resistance to be measured is placed 
between B and D, and the battery and 




Fig. 108. 



galvanometer connexions are made as shewn in the figure, the 
bridge arrangement is complete. The arms A G and BC include 
in general, resistances of 1000, 100 and 10 ohms respectively in 
<3ach branch, so that P and Q (Fig. 108) may either be equal in 
three different ways ; or in the ratio of 10 : 1 in two different 
ways ; or in the ratio of 100 : 1 by making one resistance equal 
to 1000 and the other equal to 10 ohms. According to the 
resistances placed in the arms P and Q, a resistance should 
be placed in the arm R, which, if P is equal to or greater 
than Q, should be >S', 10>Sf or 100>S, and if P is smaller than 
Q, 'IS or 'Ol^Si. Another variation can be made by inter- 



LVIII MEASUREMENT OF RESISTANCE 287 

changing the battery and galvanometer, and it is therefore 
seen that a great many combinations for producing balance 
in the bridge are available. But these combinations are not 
equally good, the unknown resistance being capable of mea- 
surement with greater accuracy for a pai'ticular choice of P 
and Q, depending on the value of the resistance to be measured, 
and on the resistances of the galvanometer and battery circuits. 
In nearly all cases in which the highest obtainable accuracy is 
not required the following rules, if attended to, will prove a 
sufficient guide to the student in the selection of a proper 
combination of resistances ^ 

(1) As the resistance R between A and i) in the ordinary 
P.O. box can be varied in steps of one ohm, and as one-tenth of 
that amount may with certainty be estimated by interpolation 
of galvanometer deflections, resistances of over 100 ohms can be 
measured with an accuracy of at least '1.7o, if the arms P and 
Q are made equal to each other. Hence if a rough measure- 
ment has shewn that 8 has a value of more than 100 ohms, 
make P and Q equal to each other. If S lies between 100 ohms 
and 10 ohms and has to be measured to less than l^o. P should 
be 10 times as great as Q, while if S has a smaller value than 
10, P should be 1000 ohms and Q 10 ohms. 

(2) When P is equal to, or ten times as great as Q, the 
best resistance of P depends on the galvanometer and battery 
resistances. For the exercises in this Section, any value will 
give sufficiently good results. It is easier in each particular 
case to vary P and Q and find the most favourable combination 
by experiment, than to determine it by calculation. 

(3) If the galvanometer has a greater resistance than the 
battery, connect the galvanometer circuit, so as to join the 
junction of the two greatest to that of the two least resistances. 
If the battery has a greater resistance than the galvanometer, 
the battery ought to be placed between the junction of the two 
greatest and that of the two least resistances. 

* A full discussion of the question is given in The tlieonj and practice of 
ahiolute meanurementt in EUctricity and MagnetUm^ by Andrew Gray, F.R.8. 



288 ELECTRICITY AND MAGNETISM LVIII 

Thus if the galvanometer has the greater resistance, and if 
P = 100, Q=10 and ^ = 8 so that iJ = 80, the galvanometer 
circuit should be connected to B and Ki, and the battery to D 
and K.2, as shewn in Fig. 108. Whenever P and Q are equal, 
the best connexion is that in which the battery and galvano- 
meter circuits are different to that shewn in Fig. 108. 

Exercise I. Measure to the nearest ohm the given re- 
sistances. The object of this exercise is to make the student 
familiar with the connexions of the P.O. Box, and to provide 
practice in determining a resistance quickly when the highest 
accuracy is not required. 

Light the scale lamp, and if the image of the wire on the 
scale is not sharp make the necessary adjustments. Bring a 
weak magnet near the galvanometer, and observe the motion 
of the spot. If the motion is smooth and regular, the needle 
swings freely. If it is irregular, adjust the galvanometer ac- 
cording to the instructions in Section LVII. Make connexions 
to the galvanometer and Leclanche cell as shewn in Fig. 108. 
Rule columns in your note-book and record the zero reading 
of the galvanometer, as indicated in the Table below. Make P 
and Q each equal to 1000 ohms. Take a resistance of 1000 
ohms out of the bridge arm AD, and to begin with, leave the 
resistance to be measured disconnected. 

Press the key K, to make the battery circuit, then for an 
instant the key K^ to make the galvanometer circuit, and 
release first K^, then K^. Observe in which direction the spot 
of light moves, and note that in the subsequent measurement a 
deflection in the same direction will always mean that S is larger 
than R ; i.e. that the resistance in the arm AB is too small. 

Now connect the resistance to be measured as shewn in the 
figure, and see that all plugs which are not taken out of the 
box are firmly in their places, and that all screw contacts are 
clean and secure. Again press down K^ and then iTj, observing 
in which direction the galvanometer begins to move. Release 
iTi quickly, so as to avoid passing a possibly large current 
through the galvanometer for a longer time than is necessary. 

If the galvanometer is deflected in the same direction as 



LVIII MEASUREMENT OF RESISTANCE 289 

before, this shews that the resistance to be measured is greater 
than 1000, because R being 1000, the deflection is in the same 
direction whether S is equal to the given resistance or infinitely 
large. In that case a higher value should be tried in the arm 
AD, say 5000, and so on until a resistance has been found which 
gives a deflection in the opposite direction. Having thus 
ascertained by trial that the resistance lies between say 1000 
and 5000 some intermediate resistance should be tried, say 
3000, and the direction of deflection will then shew between 
which limits the right value for balance lies. A succession of 
trials each time halving the resistance approximately will 
quickly reduce the limits, until two values of R are found 
differing from each other by one ohm. The one which gives 
the smallest deflection will give to the nearest ohm the correct 
resistance of S. Similarly should S have been found to be smaller 
than 1000, successive trials of 500, 250 etc. will ultimately give 
a lower limit, and when this is found a successive halving of the 
interval will again give the required resistance. 

The chief precaution to be taken by students is to avoid 
confusion as to the meaning of the defections to one side or the 
other. As soon as it has been ascertained to which side the 
deflection takes place, when the resistance taken out is too small, 
i.e. the direction of motion of the spot of light when S is not 
inserted in the circuit, this direction must be carefully and con- 
spicuously noted so that no mistake will aftei^ards be made. 

The reason for pressing down the keys K^ and K^ suc- 
cessively is to avoid effects of self-induction, which may cause 
a quite different distribution of currents at the instant of 
making the battery circuit. Pressing the key K^ when the 
connexions are as in the figure completes the circuit, and the 
first period which is affected by self-induction very quickly dies 
out, so that Kx can be pressed down almost immediately. Before 
an approximate balance has been obtained, the galvanometer 
key should only be pressed down for a sufficient time to shew in 
which direction the needle moves, as it is desirable to avoid as 
much as possible the passing of unnece-ssarily large currents 
through the galvanometer. When the balance is nearly right the 
key is kept down until the deflection can be read off either by 
8. P. 19 



290 



MAGNETISM AND ELECTRICITY 



LVIII 



noting the amplitude of the first swing or by waiting till the 
needle has come to rest. Determine the resistance of the coils 
provided and arrange your results as follows : 

20 January, 1898. 

Galvanometer No. 2 (Resistance = 67 ohms), Box C. 

P=1000. Q=1000. 



R 


Zero 


Deflected 
reading 


Deflection 


1000 

(S=QO) 


50-1 


to right 


+ 


1000 
500 
200 
400 
450 
420 
410 
412 
413 
411 


501 
50-1 


to left 

>> 
to right 
51-4 
46-2 
49-2 
50-3 
50-1 
50-0 
50-2 


+ 
+ 1-3 
-3-9 

- -9 
+ -2 
zero 

- -1 
+ -1 


.S' = i? = 412ohms. 



Until a student has obtained sufficient practice, his Book 
of Observations should contain a statement of all his observa- 
tions as above. In writing out his results one example should 
be given in full ; for the other resistances measured, the results 
only need be given. 

Exercise II. Determine the same resistances to the 
greatest accuracy which the apparatus at your disposal will 
allow. 

The resistances having been approximately determined, the 
student should for each of them separately consider the rules 
mentioned at the beginning of this section and fix on the values 
of P and Q which he considers most suitable. The rule about 
the proper connexion of galvanometer and battery should also 



LVIII 



MEASUREMENT OF RESISTANCE 



291 



be attended to. When the galvanometer and battery circuits 
are interchanged, the order in which the keys K^ and if, are 
pressed down must of course be reversed also. From his 
previous results he will at once be able to take out of the arm 
AD the plugs necessary to adjust the balance almost correctly. 
A few further trials, if necessary, will then again lead him to 
the two values R and ii + 1 between which the balance lies. 
The deflections for both these must then be carefully observed 
on the scale, and by interpolation one, and possibly two 
decimals may be found. 

If the deflections are not sufficiently large, the galvanometer 
must be made more sensitive. Resistances of not less than 
five ohms should in each case be accurately determined to -1 */„ 
and if possible more accurately. To ensure this accuracy, 
however, with the smaller resistances it will be necessary to 
attend carefully to the connexions and to make sure that the 
plugs in the resistance box are firmly in their sockets. 

If the resistances are of copper or some other unalloyed 
metal, the resistance of which increases about 1 7o for every 
3° C, the temperature should be noted, and the current should 
be passed through the coil for as short a time as possible, so as 
to avoid heating effects, otherwise the resistance will be found 
to increase gradually. To make sure that no such change is 
taking place, the observations should be repeated. Owing to 
the temperature effect, the accuracy obtainable with pure 
metals is much smaller than with alloys, which have a small 
temperature coefficient. 

Enter your results as follows : 





Material 


P 


Q 


Junction between 

Pand Q 

connected to 


R 


Temp, 
of coil 


Coil No. 1 
2 


Copper 

Platinoid 

Manganin 


1000 
100 

100 


1000 
10 

10 


Galvanometer 
Battery 

Battery 


412-3 
68-25 

7-423 


16'-2 
16'-4 

16--9 



19—2 



292 MAGNETISM AND ELECTRICITY LVIII 

Exercise III. To determine the resistivity of the material 
of the wire. 

Measure the resistance of the wire provided to 'lYo, 
noting the length of the wire under the screws which clamp 
it to the resistance box. Measure the diameter of the wire in 
four places and the total length. Subtract the length under 
the clamping screws from the total length, and calculate the 
resistivity. 

Arrange your results as follows : 

21 January, 1898. 
Resistivity of Platinoid. 



Total length of wire 




= 100-2 cms. 


Length under screws 


... 


= 1-7 „ 


Length (I) used 


... 


= 98-5 „ 


Diameters at different places 

screw gauge 
Mean diameter ... 


by. 


= -0.342, -0344, -03- 
= -0345 cms. 


Mean cross section (a) ... 


... 


= -000937 sq. cms. 


Resistance (R) 


... 


= 4-205 ohms. 


Resistivity = -r- 


... 


= 40-0 X 10-« ohm 



SECTION LIX. 

MEASUREMENT OF A HIGH RESISTANCE. 

Apparatus required : High resistances, resistance boxes, 
cells and mirror galvanometer. 

When the resistance to be measured, is large compared 
to that of the most sensitive available galvanometer, the 
Wheatstone bridge loses its advantages and ceases to give 
more accurate results than simpler and more direct methods. 
When the accuracy required does not exceed about *! 7o these 
simpler methods will be sufficient. 

If a standard resistance of approximately the same value as 
the one to be measured is available, the method known as the 
" direct deflection method " may be used. 

Let a cell of resistance B ohms be connected in series 
with a high resistance of R^ ohms, and a 
mirror galvanometer of resistance G ohms 
(Fig. 109). 

The current C^ through the circuit is 
given by 

C- E 

Similarly when a second known resistance R, is substituted 
forR,, 

C- g 

so that 

B + + R,-^'(B + + R,). 




294 MAGNETISM AND ELECTRICITY LIX 

The battery resistance will in nearly all cases be quite 
negligi\)le compared to the resistance to be measured, and the 
resistance of the galvan<>meter will be known or must be 
determined by an independent measurement. Hence R^ being 
known, R^ can be calculated from the above equation, if the 
deflections of the galvanometer d^ and d^ corresponding to Ci 
and Cg are observed, and the ratio of currents deduced from 
the ratio of deflections. If the indications of the mirror 
galvanometer have been calibrated (page 281) a Table is 
probably available in the laboratory by means of which de- 
flections may be converted into currents. In the following 
exercise it is assumed that results of sufficient accuracy may 
be obtained by assuming the deflections to be proportional to 
the currents. In that case the unknown resistance R^ is 
given by 

and if G is small compared to R^ and R2 

Taking a resistance box having a total resistance of not less 
than 10000 ohms as R^, determine the re- 
sistance Rz of the coil provided. Arrange the . — (^ 
circuit as in Fig. 110, the resistances to be 
compared being placed side by side in such a 
way, that either one or the other may be put in ' .' 

circuit with the battery and galvanometer. 

If the deflections of the galvanometer are too large to enable 
observations to be taken, diminish the sensitiveness of the 
instrument by an external magnet. If this is not possible 
arrange the circuit as in the second method described below. 

If the resistance to be determined is more than three or four 
times that of the standard, the ratio of the currents will not 
be capable of measurement to a sufficient degree of accuracy by 
the direct deflections. The method should then be modified 
by the addition of an arrangement which allows the electro- 



_Lb 



LIX MEASUREMENT OF A HIGH RESISTANCE 295 

motive force to be varied in such a way that the deflections 
di and (L are not very unequal. 

For this purpose a cell is connected to the ends of a 
resistance box containing about 1000 ohms. 
The galvanometer with high resistance to 
be measured, is connected to part of the 
resistances only (Fig. Ill), and this part 
may be varied. Ivfy 

Let the resistance to be measured be R^ — lisp-v 

and the standard resistance ii,, and write M fi, H^ 

S,^R, + G, S, = R,-¥G, Fig. 111. 

where G is the galvanometer resistance. 

Let r be the total resistance inserted in the box, and ri, r^ 
that part of it, which lies between the terminals of the galva- 
nometer circuit according as R^ or R^ is included, the currents 
through the galvanometer being in these cases Ci or C^ ; then if 
E is the electromotive force of the battery, the resistance of 
which may be neglected compared to the other resistances in 
the circuit, a simple application of the laws of derived circuits 
gives 

C,(^,r + (r-rOn) = AV„ 

C, {S^r + (?• - ra) ra) = Er^, 

from which we find 

C^r,\ r J r 

Q 

Hence R^ may be determined, if we can tind the ratio 77 

from the galvanometer deflections. If the currents are not pro- 
portional to the deflections or to their tangents, the galvanometer 
must be calibrated. But this can be avoided and the arith- 
metical work shortened if the resistances r,, r^ are so adjusted 
that the deflections are equal. In that case 

The above equation also simplifies if the total resifltftDoe is 



296 



MAGNETISM AND ELECTRICITY 



LIX 



inserted, when the larger of the two resistances R^ and R^ is 
in circuit. In that case 



when 


r^=-r, 


we have 


^-S(¥-^(--)) 


and when 


7-1 = r, 


we have 





Should it be necessary to take the battery resistance into 
account, we need only add its value to that of r in the complete 
equation. 

Determine by this method the resistances of the blacklead 
line, and of the samples of insulating material provided. From 
the dimensions of the samples, calculate the resistivities of 
the different materials. 

Assuming the deflections to be proportional to the currents, 
tabulate your observations and results as follows : 



Direct Deflection Method. 

20 January, 1898. 

Galvanometer No. 2 (Resistance 6,430 ohms). 

Standard Resistance (R^) = 10,000 ohms. 



Besistance 


Deflection 


10000 
10000 


508-7 
274-5 
510-3 


d,= 509-5 
d,= 274-5 



R, = 16430 X 2^^ - 6430 = 24,066. 



LIX MEASUREMENT OF A HIGH RESISTANCE 

Shunt Method. 

20 January, 1898. 

Galvanometer No. 2 (Resistance 6,430 ohms). 

Standard Resistance = 25,000 ohms. 

r= 11,000 „ 



297 



Besbtanoe 


ri 


Deflection 


25000 

Pencil Line 

25000 


1000 

11000 

1000 


430 
489 
436 


rf, = 433 ri= lOuO 6', = 31430 
d^ = iS9 r, = r= 11000 



4,00 

S, = *^^ {(31430 X 11) + 10000) 

= 314,900, 
222=208,500 ohms. 

Similarly for the samples of insulating material. 



SECTION LX. 

MEASUREMENT OF A LOW RESISTANCE. 

Apparatus required : Low resistances and standard ad- 
justable resistance^ Daniell cell, two four-way keys, and mirror 
galvanometer. 

When a resistance which is only a small fraction of an ohm 
is to be measured, the Wheatstone bridge is no longer capable 
of giving accurate results owing to the resistances introduced at 
the various contacts. Several special methods have been devised, 
one of which is known as the *' fall of potential method." It 
depends on the comparison of the differences of potential 
produced by the same current at the ends of the unknown 
resistance and of a known standard resistance. 

The two resistances R^ and R.^ (Fig. 112) are connected in 
series with a third resistance i2 of a few- 
ohms, introduced to regulate the current, 
and to a cell B giving a constant electro- 
motive force, e.g., a Daniell. From the 
ends of the two resistances R^ and R^, 
wires are taken to keys which enable each 
resistance to be placed in parallel with a 
galvanometer G, the resistance of which is ^10,. 112. 

high compared to the resistances under 
test. When this is the case, the currents in the galvanometer 
will be proportional to the differences of potential at the ends 
of the resistances, which will in their turn be proportional to 
the resistances themselves. The galvanometer should give de- 
flections sensibly proportional to the currents passing through 
them, or if this is not the case, its indications should have been 
previously calibrated. 




LX 



MEASUREMENT OF A LOW RESISTANCE 



299 



If i^a is a standard resistance not many times greater or 
many times smaller than i^, the latter may be determined by 
this method with an accuracy which is generally sufficient. 

The connexions to tiie galvanometer may be arranged by 
means of two four- way keys, K^ and K^ as indicated in the 
figure, the plugs being inserted to connect the galvanometer to 
the two resistances in turn. 

After making connexions proceed as follows : 

Make i2=10 ohms, connect the galvanometer to R^, and 
read the deflection. If the detlection is so great that the spot 
of light leaves the scale, increase R. 

Connect the galvanometer to ilj and again read the 
deflection. 

If the larger of the two deflections is not at least half way 
across the scale, connect the galvanometer to the resistance 
which gives the larger deflection, and diminish R till this 
deflection is as large as can conveniently be read. Determine 
it accurately. Transfer the galvanometer terminals to the 
other resistance and read the deflection, then return to the 
former and again read the deflection. If any measurable 
change has occurred take two more observations and record as 
shewn below. In the reductions, assume that the galvanometer 
deflections are proportional to the currents. 

Enter your observations as follows : 

10 June, 1899. 
Galvanometer No. 3. Resistances marked A, B, G. 



Galv. connected 
up to 


Deflections, 
scale divisions 


Mean 

114-5 
29-6 


^.W 

J?, = 01 ohm 


114 


... 115 
29-4 ... 


29-8 


114-5 



H( 



R, 



114-5 



= ;}-87 



29-6 

or resistance A = '0387 ohm. 

Similarly for the other given resistances. 



SECTION LXI. 

MEASUREMENT OF THE RESISTANCE OF A 
GALVANOMETER. 

KELVIN'S METHOD. 

Apparatus required : Galvanometer, Post Office resistance 
box, Leclanche cell, resistance box of 1000 ohms and connecting 
wires. 

In the ordinary Wheatstone bridge arrangement, if the four 
arms of the bridge (Fig. 108, p. 285) satisfy the relation 
F/Q = R/S there is no difference of potential between the 
points A and B, and consequently if these points be connected 
by a wire, no current will pass through it. Such a wire would 
therefore not alter the strength of the currents in any of the 
branches of the bridge and on the other hand we may conclude 
that the above relation is satisfied, if the current in any one 
of the branches is the same whether A and B are directly 
connected together or not. If the branch R contains a 
galvanometer, the deflection of which serves to indicate the 
strength of the current through it, we may judge whether the 
resistances are balanced or not by making and breaking the 
contact of a wire connecting A and B. If the deflection is the 
same in both cases the resistance in the branch R must be 
equal to PS/Q and may therefore be determined without the 
assistance of a second galvanometer between A and B. 

The above considerations furnish an interesting and useful 
method to determine the resistance of a galvanometer. To 
obtain accurate results it is in general necessary to send a 
current through the instrument, which would under ordinary 
circumstances drive the spot of light off the scale. Reducing 



LXI RESISTANCE OF A GALVANOMETER 301 

the sensitiveness of the galvanometer so as to make the 
deflection measurable would not get over the difficulty, because 
the test itself would become less sensitive. It is possible 
however to work with a large deflection and yet have the spot 
of light on the scale, because the zero reading of the galvano- 
meter is not required and may therefore be outside the limits 
of the scale. In the case of a galvanometer of the suspended 
needle type it is convenient to begin by deflecting the needle 
with the help of a weak magnet, which should turn the field 
rather than increase its strength, and the deflection should 
not be too large to begin with, so as to save passing currents 
through the galvanometer, which may cause damage. If the 
battery is then connected through a variable resistance, as 
shewn in Fig. 118, so that only a portion of the current passes 
through the bridge, the spot of light is brought to the scale by 
variation of the resistances r and r^. Care is necessary of course 
to send the current in the direction necessary to bring the 
suspended system into the required position. 

The method adapts itself very easily to the measurement of 
the resistance of a D'Arsonval galvanometer, for it will only be 
necessary to rotate the whole instrument through an angle of 
about 30° and to adjust r and 1\, until the deflection can be 
read on the scale. 

The best arrangement of the bridge, when Q may be made 
smaller than the resistance of the galvanometer, is that in 
which Q is as small as possible and P as large as possible 
without R being caused to exceed the maximum resistance 
available in the arm AD. The battery should connect the 
points A and By i.e., the battery and short circuiting wire in 
the figure should be interchanged. When the lowest available 
resistance of Q is larger than the galvanometer resistance, Q 
and P should be made equal to each other and as large as 
possible, and the battery should then be connected as shewn in 
the figure. If the resistance of the galvanometer is not even 
approximately known the last-mentioned arrangement is the 
most suitable and the sensitiveness will generally be sufficient 

In the following exercise, it is required to measure the 
resistance of a d'Arsonval galvanometer, the method for other 



302 



MAGNETISM AND ELKCTRICITF 



LXI 




Fig. 113. 



galvanometers only differing as explained above in the means 
adopted to alter the zero reading. 

Make connexions as shewn in the figure. With rj small 
and r large, pass a current of short duration 
through the galvanometer by pressing down 
the key K^ instantaneously. 

Turn the galvanometer through an 
angle of about 30" or 40° in the opposite 
direction to that in which the spot of light 
moved on making contact. 

With F=^Q = R = 1000 close the bat- 
tery circuit at the key K^ and adjust 7\ 
and r until the spot of light is on the 
scale. This is done most rapidly by 
watching the galvanometer mirror directly. 
Press the short circuiting key K^ and observe the side to which 
the spot of light moves. Then make R = 5000 and repeat the 
observation. If the motion is in the same direction as the first 
it indicates that S lies below 1000 or above 5000. The first 
supposition being the more probable in .the present case, repeat 
the observation with i^ = 500. 

Proceed as in Section LVIII to find the value of B for 
which the spot of light does not move when the key K^ is closed. 

When the first deflection of the needle is sufficiently small, 
take readings of the first swing and find the value of R for 
accurate balance, if necessary by interpolation. 

If the resistance of the galvanometer is greater than the 
lowest available resistance of Q, which will be 10 with the 
resistance boxes in common use, interchange the battery and 
short circuiting wire. Make Q=10 and P = 1000 if a resistance 
of 100 >S^ can be placed in the arm AD, otherwise make P= 100. 
Take out of the branch AD the resistance which according to 
the previous determination should produce balance; and if 
there is a deflection proceed to improve your result. When 
the balance is nearly perfect wait a few minutes in order to 
allow the galvanometer coil to take up the temperature of the 
room, it having probably been heated by the passage of currents 
sufficiently to affect its resistance sensibly. 



LXI RESISTANCE OF A GALVANOMETER 

Enter your results as follows : 

9 June, 1899. 

Galvanometer D' Arson val C. 

Battery joining C and D. 

P = 1000, Q=1000. 



303 



R 


Beading with 
JBopen 


Beading with 
AB closed 


Deflection 


1000 


15-1 


+ 


large to right 


5000 


121 


+ 


jj » 


500 


23-2 


+ 


j> >> 


100 


25-4 


— 


„ to left 1 


300 


240 


41-3 


+ 17-3 


200 


240 


22-5 


- 1-5 


220 


23-6 


24-5 


+ -9 


210 


23-4 


22-4 


- 10 



Resistance of galvanometer = 215 app. 
Temperature of room = 15°'4. 

The battery and short circuit were now interchanged and 
with Q=10, P = 100 it was found that P = 2154 gave a 
balance but that this resistance seemed slowly to increase 
owing no doubt to the heating of the current. The same 
result was obtained after waiting five minutes. Hence for 
the final result : 

Resistance of galvanometer = 215*4 
Temperature = 15°'4. 



SECTION LXII. 

DETERMINATION OF THE RESISTANCE OF A CELL. 
lodge's modification of mange's method. 

Apparatus required : Daniell and LeclanchS cells, Post 
Office resistance box, high resistance galvanometer. 

If in the Wheatstone bridge arrangements (Fig. 108, p. 285) 
the relation PS = QR is satisfied, the portions of the circuit CD 
and AB are said to be conjugate to each other. No electro- 
motive force in one of these portions will produce a current in 
the other, and no change of resistance in one of them will 
modify any current in the other, such 
as may be produced by an electromotive 
force in one of the branches of the bridge. 
This fact was first used by Mance to 
determine the resistance of a cell. He 
placed the cell in one of the branches, 
and replaced the battery which in the 
ordinary Wheatstone bridge supplies the 
current, by a simple key iL'2(Fig. 114). 
Owing to the presence of the cell in BD Fig. 114. 

a current passes through the galvano- 
meter, but when PjQ — R/S this current should be the same 
whether the contact at K^ is open or closed. 

Mance's method in its original form has several disadvan- 
tages. A comparatively large current has to be sent through 
the galvanometer and the key K^ has to be pressed down, at 
least for the length of time equal to one quarter of the period 
of the galvanometer suspension, so as to allow the first deflection 
to be read. This may affect the galvanometer injuriously, and 
it also produces disturbing effects in the cell itself The closing 
of the circuit at K^, though it does not affect the current in the 




LXII DETERMINATION OF THE RESISTANCE OF A CELL 305 

galvanometer branch, will increase the current through the cell 
itself, and this increase is accompanied by changes in the resist- 
ance and electromotive force of the cell. This is a disturbing 
cause which affects all measurements of the resistance of a cell, 
but the difficulty may to some extent be overcome by allowing 
the key K^ to be closed only for a very short interval of time. 

The modification of Mance's method, introduced by Lodge, 
gets over the difficulty by the introduction of a condenser in 
series with the galvanometer (Fig. 115). The condenser will be 
charged to a difference of potential equal to that between the 
points A and B\ and whenever a sudden change in that 
difference of potential occurs, an instantaneous current will 
pass through the condenser. When however the relation 
PjQ — RIS is satisfied, the galvanometer will not be affected 
by the sudden pressing down of the key K^. 

The most sensitive arrangement for the determination of 
the resistance of a cell is that, in which Q is equal to the 
resistance of the cell and P is as large as possible. The 
connexions should be as in Fig. 115, the short circuiting key 
being in the branch CD. It is more important that Q should 
be nearly equal to the cell resistance, than that P should be 
large. If P = lOQ the arrangement has 90 7© of the greatest 
possible sensitiveness, but when Q=^\OS the sensitiveness is 
reduced in the ratio of 10 to 1. Hence when the cell has a 
very low resistance, as e.g. a storage cell, either some special 
arrangement should be made to reduce the value of Q, 
or an accurately known resistance should be placed in series 
with the cell, so that the total resistance 
of *S' is not less than half that of the 
lowest available value of Q. 

Set up a Daniell cell, taking care 
that the zinc is not covered with a 
deposit of copper. Place it in the arm 
BD of the Wheatstone Bridge Box 
ABCD (Fig. 115), putting a plug key 
K in circuit, in order that the current 
can be stopped except when observations 
are being taken. 

8. p. 20 




306 MAGNETISM AND ELECTRICITY LXII 

Place a condenser T of about "3 microfarad capacity in series 
with a high resistance galvanometer, and connect to the bridge 
as shewn. Make Q=10, P = 1000, and choose for first trial a 
resistance R which you expect to be greater than that of the 
cell to be tested, viz. 100 in the present instance. Make circuit 
at K^y then momentarily at K^. Note the direction in which 
the spot of light moves. Reduce the value of R and proceed 
as in Section LVIII. to determine its value for balance. If the 
key K. is held down, the spot of light will be seen to drift owing 
to changes in the cell, brought about by the passage of the 
current. Wait five minutes without passing a current through 
the cell, and determine the resistance again. Repeat again 
after further five minutes. 

Determine the resistance of the Leclanche cell in the same 
way, then placing the two cells first in series, then in parallel, 
determine their joint resistance. 

Record as follows : 

6 February, 1899. 
P=1000, Q= 10 ohms. 

1. Daniell cell, No. 3. 

Balance obtained with R = 515, i.e. S = 5 15 ohms. 
After five minutes ... ... ... =5*10 „ 

After further five minutes ... ... =5*07 „ 

2. Leclanche cell, No. 4. 
Approximate balance obtained for R 

between 470 and 480 S = 4*75 approx. 

After five minutes and after a further 
five minutes, the resistance was 
found approximately the same. 

The resistance therefore varies be- 
tween 4*7 and 48 ohms. 

3. Leclanche and Daniell cells. 

(a) in series ... ... ... ... S = 8'9 ohms. 

(6) in parallel =22 „ 

Calculated resistance case (a) ... = 98 „ 

case (6) ... =25 „ 



SECTION LXIII. 



COMPARISON OF RESISTANCE STANDARDS. 



CAREY FOSTERS METHOD. 



Apparatus required : Carey Fosters bridge with sioitch, 
two equal resistances, standai^ds to be compared^ miiTor galvano- 
meter, voltaic cell. 

When nearly equal resistances, as for instance a number 
of ohm standards, are to be compared together with gi-eat 
accui-acy, a modification of the bridge method due to Carey 
Foster may be used. 

The two "proportional arms" P, Q (Fig. 116) of the bridge 
are made as nearly as possible equal, and 
the two standards R, S to be compared 
together with a straight wire CC join- 
ing them form the other arms. The 
standards R, S are connected to the 
bridge in such a way that they can be 
readily interchanged by means of a 
switch having mercury contacts. The 
wire CC is stretched along a gradu- 
ated scale, on which the position of the 

point of contact of the sliding connexion leading to the 
galvanometer may be read off. 

Let a balance be found when the galvanometer contact is 
made at a point reading x along CC\ R and S being placed 
as shewn in the figure, and let x' be the position for a balance 
when R and 8 are interchanged. Let p be the resistance 

20—2 




308 MAGNETISM AND ELECTRICITY LXIII 

of a length of wire GC equal to one scale division, then since 
the resistance from B to D through CC remains the same on 
interchanging R and S, the sum of the resistances BL and LD 
remain the same ; and since when there is a balance their ratio 
must also be the same, it follows that the resistance from Bio L 
is the same before and after the change. 

Hence R + px = S-\-px 

or R — 8 = p(x— x). 

If p is known the difference between R and S can therefore 
be calculated. To obtain p diminish the effective resistance of 
S by placing a large known resistance Si in parallel with it. 
Balance, interchange the two arms, and balance again. Let x^ 
and Xi be the two readings. 

Then 

~ SVS ^P^^' ~ ^'^' 
Eliminating R we obtain 

^_S^ 1 

S + Si (xi — Xi) — (x — x) ' 

It is here assumed that the bridge wire is uniform, so that 
its resistance is strictly proportional to its length. When the 
highest accuracy is required, the wire should be calibrated so as 
to correct for any inequality either of the cross-section or the 
material of the wire. 

See that the mercury contacts of the switch are clean. 
Place the terminals of the two equal coils, which are best wound 
on the same bobbin, and those of the two coils to be compared 
in the mercury cups provided for them in the mercury switch. 
Connect the Leclanch^ cell through a key, and the galvanometer 
to the bridge, attending to rule (3), page 287. Place a thermo- 
meter in the centre tube of each coil, and when the indications 
have become steady, read the temperatures. Obtain a balance 
and note the reading of the sliding contact. Reverse the con- 
nexions of the standard coils by moving the switch, balance 
again and note the reading. Read the thermometer again. 
Place, in parallel with S, a high resistance of amount sufficient 



LXIIl COMPARISON OF RESISTANCE STANDARDS 309 

to displace the point of balance by about a fourth of the length 
of the wire, and determine the position for balance. Again 
reverse and balance. Read the temperatures. 
Record as follows : 

2 October, 1900. 

Wolff's standard 1 ohm (R) compared with Hartmann and 
Braun's standard 1 ohm (S). 

Temperature coefficient of R said to be zero 
,, ^=-00020 
S according to the certificate is correct at 18'''2. 

Balance with R on left side of bridge ... x = 24*38 cms. 
„ „ o „ „ „ „ .,. X = ^4"7o „ 

.-. R^S='S7p, t = 15°'lo. 
100 ohms were placed in parallel with S and difference in 
readings now found to be 4*52 cms. 

Hence p^^^.^^^-^ 

= ^^ A^ = 00239 ohms per cm. 
419'2 ^ 

.-. R-S^002S9 X -37 = -00088 ohms. 

Resistance of S at 15°-2 = 1 - 00060 = -99940 ohms, 

of ie at 1 5°-2 = 1 00028 ohms. 



SECTION LXIY. 

CHANGE OF ELECTRICAL RESISTANCE WITH 
TEMPERATURE. 

Apparatus required : Two coils of luire in a tube which 
can be raised in temperature, Post Office bridge, voltaic cell, 
and mirror galvanometer. 

The electrical resistance of a wire of a pure metal increases 
rapidly with increase of temperature, while that of a wire of an 
alloy increases more slowly, and by a proper choice of the con- 
stituents, may be made to remain nearly constant at ordinary 
temperatures. 

To determine the change of resistance of a wire due to 
temperature, the wire may be wound round a sheet of mica 
and placed, along with a thermometer for indicating its tem- 
perature, in a test tube surrounded by water which can be 
heated. 

The apparatus supplied is constructed on this principle. 
It consists of a brass vessel, in which is placed a test tube 
containing two coils, one of copper and one of platinoid wire, 
immersed in petroleum and joined to the three screws on the 
wooden disc through which the tube passes, in such a way that 
the middle screw is connected to one end of both coils, while 
the other ends of the coils are separately connected to the 
other two screws. 

Fill the brass vessel full of tap water, and place the disc 
through which the test tube passes over the vessel, so that 
the tube is immersed in the water (Fig. 117). Connect the 
common terminal of the two coils to a Post Office Wheatstone 



LXIV 



CHANGE OF RESISTANCE WITH TEMPERATURE 



311 



bridge in the usual way, and the other terminals to two of 
the screws of a three-way key, the third screw 
of which is connected to the other bridge ter- 
minal, and determine the resistance of each 
coil at the temperature of the bath. 

The galvanometer should be sufficiently 
sensitive to allow the thousandth part of an 
ohm to be estimated, and the temperature 
readings should be correct to '05° C. 

Rixise the temperature of the water to 
about 20° C, and keep it at that temperature 
till the reading of the thermometer in the 
test tube is steady, then repeat the obser- 
vations of the resistances of the two coils. 

Take further observations at about 30°, 
40°, 50°, 60°, then cool the water, and redeter- 
mine the resistances at the same temperatures 
(within about a degree) as previously. 

Record as follows: 

12 December, 1898. 
Post Office Box C. Coils No. 3. 




Fig. 117. 



Copper CoU. 


Platinoid CoU. 


Temp. 
C. 


Resist, 
olims 


' Temp. 
1 C. 

i 


Resist, 
obms 


Temp. 


Resist. 


Temp. 


Resist. 


13'-62 
20 15 
29-80 
39 15 
49-50 
59 -90 


4'-168 
4 -274 
4 -418 
4 -552 
4-727 
4-915 


13" -58 
20 05 
29 -90 
39 -35 
49 -60 
1 59 -70 


1 

4162 
4-258 
4-434 
4-584 
4-743 
4-881 j 











Draw curves representing the results, taking temperatures 
08 abscissae and resistances as ordinates. 

It will be found that for the copper coil the curve is almost 
a straight line, which, when continued so as to take in tem- 
peratures below the freezing point, gives zero resistance at the 
zero of absolute temperature. 



312 MAGNETISM AND ELECTRICITY LXIV 

All pure metals behave in the same way but the resistance 
of alloys follows other laws. 

The resistance of a pure metal may be represented very 
accurately by the equation 

?V = ?'o(l + a^ + /3«'), 

where Vt is the resistance at f and r^ at 0°, and a and /3 are 
constants depending on the nature and state of the metal. 

The temperature coefficient of increase of electrical re- 
sistance at f is the increase in resistance of a conductor for a 
rise of temperature from {t — \y to (^ + i)° divided by the 
resistance of the conductor at 0°C. 

It is therefore in the above case a + ^(St 

The quantity y8 is small compared to a : thus for platinum 

a = + 003448, yS = - 0-000000533. 

For many purposes it will be sufficient to neglect /9, and to 
take a to be the mean temperature coefficient for the range of 
temperature considered. In that case, writing R^ and R.^ for 
the resistances at the temperatures t^ and 4, we have 

R^^r,{\+at,), 

R^ = r,{l+OLU), 

and by elimination of r^ 

!._ Jl 

Xt2 — -til -^1 R^ 

jRa Ri 

The first formula involves rather less calculation than the 
second, but the second allows a more symmetrical arrangement 
of the observations and calculations, and as tables of reciprocals 
should be available in every laboratory, the introduction of con- 
ductances in place of resistances involves little additional labour. 

Calculate the temperature coefficient of the electrical re- 
sistance of copper by taking the mean of each pair of nearly 
equal temperatures in the first and third columns of the above 
Table and the mean of the resistances found for these tempera- 
tures. Arrange them in two sets and carry out the work as follows: 



LXIV CHANGE OF RESISTANCE WITH TEMPERATURE 

12 December, 1898. 
Copper. Coil 3. 



313 



MeM 


Mean 
re«Ut. 
Ohms. 


1 
R 


( 
^ 


Meui 
temp. 
Cent. 


Mem 
reebt 
Ohms. 


1 
R 


t 
R 


Diff. of 

1 
R 


Diff. of 

t 
R 


lUtio 


13-60° 

2010 

29-85 


4-165 
4-266 
4-426 


•2401 
•2344 
•2259 


3-265 
4-711 
6-744 


39-25° 

49-55 

59-80 


4-568 
4-735 
4-898 


•2189 
•2112 
-2042 


8-591 
10-466 
12-211 


•0212 
-0282 
•0217 


5-326 
5-755 
6-467 


•00398 
403 
397 


















a = mean = 


•00399 



Reduce the results for the platinoid coil in the same way 
and examine whether in its case the above simple relation 
between temperature and resistance holds. 

Note on Platinum Thermometry. The measurement of 
change of resistance due to change of temperature has become 
of considerable importance, since it has been found that it may 
serve as a basis for the measurement of temperature. One of the 
advantages of this method is, that by the use of a metal with a 
high fusing point, e.g. platinum, it may be applied to tempera- 
tures which are so high that mercury or air thermometers 
cannot be used. A platinum thermometer must agree with 
the ordinary scale at the fundamental points 0° C. and 100° C, 
and this may be done by defining the temperature as measured 
by the platinum thermometer {tp) to be 



^^=100-?---^ 



where r,, r^, r,oo are the resistances of a platinum wire at 
0°, p", and lOO"* respectively. This definition makes the rise of 
temperature proportional to the increase in the resistance, as is 
seen by writing the difference between two temperatures tp and 



tq in the form 



<p-<,= 100 



^ifflo ~~' rn 



The definition secures also that when rp is equal r^ or Tim the 
temperature indicated shall be or 100 respectively. 



314 MAGNETISM AND ELECTRICITY LXIV 

The difference between a platinum and an air thermometer 
at any temperature can be calculated, if the change of resist- 
ance of platinum as depending on the indications of the air 
thermometer is known. Thus putting 

rp = ro(l+a« + W, 

where t is the temperature as measured in an air thermometer, 
we find by substitution 

^ en- 100^ 

This equation was first used by H. L. Callendar, who has 
measured the quantities a and /8 for pure platinum with great 
care and found the values given, p. 312. With these values 
the difference of the two temperature scales becomes finally 

^-.^^ = -0-157^ (100-0. 

The numerical coefficient has however to be redetermined for 
each sample of platinum. 



SECTION LXV. 

MEASUREMENT OF THE RESISTANCE 
OF ELECTROLYTES. 

Apparatus required : Post Office resistance box, electrolyte 
tube, telephone, induction coil, storage cell, and microscope. 

The resistance of an electrolyte cannot be measured by 
placing the vessel containing the liquid in the Wheatstone 
bridge in the ordinary way, since the passage of the current 
produces polarisation at the electrodes, and therefore sets up an 
electromotive force, which has the same effect on the measuring 
instrument as a change of resistance. If however an alternating 
instead of a direct current is sent through the bridge, the 
polarisation due to the passage of the current in one direction, 
is neutralised by the passage the next instant of an equal 
current in the opposite direction, and if the changes succeed 
each other with sufficient rapidity no appreciable effect on the 
measurement is produced. The galvanometer must however 
be replaced by an instrument capable of detecting alternating 
currents, as for example an electrodynamometer or a telephone. 
The latter is more generally used at present. The use of 
alternating currents necessitates care in avoiding appreciable 
self-induction and capacity in the resistances, since the Wheat- 
stone bridge does not measure in that case "resistance" but 
" impedance," which depends on the self-induction, the capacity 
and the number of alternations per second as well 8is on the 
resistance. It is owing to the impossibility of completely 
getting rid of or balancing self-induction and capacity, that 
no perfect balance is ever obtained unless the alternating 
currents used follow the law of sines. 



316 MAGNETISM AND ELECTRICITY LXV 

Important conclusions on the molecular constitution of 
solutions have been drawn from their electric conductivities, 
the solutions of different substances being compared when their 
concentration is such, that an equal number of molecules are 
dissolved in each case. If the number of atoms in one gram of 
hydrogen be 91, and a represent the molecular weight of a 
substance referred to hydrogen, then a grams of any substance 
will contain the same number n of molecules, and different 
solutions each containing a grams per litre will therefore be 
comparable with each other. The mass of a substance which 
contains as many grams as is indicated by its molecular weight, 
is called a " gram-molecule," an expression which, it is hoped, 
will before long be replaced by a more appropriate one. A 
solution which contains one gram-molecule of a salt per litre of 
water is called a "normal solution." The molecular conductivity 
of an electrolyte is the conductivity divided by the number of 
gram-molecules which are dissolved in one cubic centimetre. 
If it is required to measure the conductivity, not only relatively 
to some standard solution but absolutely, the electrolyte must 
be inclosed in a tube of known length (I) and cross-section (a). 
If R is the resistance of such a tube Ra/l will be the resistivity, 
and l/aR the conductivity. If the tube is conical, a^ and Og being 
the cross-sections at the ends, it will have a resistance equal to 
that of a uniform tube having a cross-section equal to s/a^a^. 
If each cross -section is circular, it is calculated in the usual way 
from the diameter, but if it is elliptical in shape, the two prin- 
cipal diameters d^, d^ must be measured, and the area is then 
equal to ^irdyd^. If d^, d4 represent the principal diameters at 
the other end, the cross-section of the equivalent uniform tube 
will therefore be \'Tr\fd^d^Qd^. In general the four diameters 
will be nearly equal, and if they do not differ by more than 2 or 
3 per cent, it will be sufficient to substitute arithmetical for 
geometrical means, so that ii D = \{d^ + d^-\- d^ + d^) the area to 
be used in the reduction of the experiment will be ^irD'^. 

Wash out the narrow glass tube and wider end tubes 
provided, and fill them with a solution of sodium chloride 
of four times normal strength, i.e. containing 4 gram-molecules 
per litre, place the platinum electrodes in the end tubes and 



LXV 



RESISTANCE OF ELECTROLYTES 



817 



connect to the bridge as shewn in Fig. 118, taking out the 1000 
ohm plugs from each arm of the box before connecting up. 




Fig. 118. 

Place the bulb of a thermometer in one of the end tubes, taking 
care that it does not come between the platinum electrode and 
the end of the narrow tube. To the points C, D of the bridge 
to which the battery is usually connected, join the terminals of 
the secondary of a small induction coil /, the primary of 
which is excited by a cell E of sufficient power to work the 
coil. Connect a telephone T to the terminals A, B. 

Determine the resistance in the adjustable arm of the 
bridge to produce a minimum sound in the telephone. 

Take about 50 c.c. of the solution and dilute to double the 
volume, i.e. make a solution of twice normal strength, and after 
washing out the tube, fill it with the new solution and deter- 
mine the resistance. Dilute down to normal, then to half, 
and quarter normal strength, determining the resistance and 
observing the temperature in each aise. 

Remove the end tubes of the electrolyte cell, and measure 
the internal diameter of each end under the microscope. If 
the tube is not quite circular measure the least and greatest 
diameters of each end. Measure the length I of the tube. 



318 MAGNETISM AND ELECTRICITY LXV 

Arrange your observations and results as follows : 

12 December, 1898. 
Electrolyte tube A. Resistance box C. Microscope A. 
75*4 eyepiece divisions = 1 cm. of stage scale, 
.-.1 „ „ =-0133 cm. 

Diameters = 236, 232, 234, 234 eyepiece divisions. 
Mean Diameter = 23*4 eyepiece divisions, 



= •311 cms. 

Area = ?^. (-311)2 

Length of tube Z = 129 cms. 
a 



•0759. 



= •OOoSS. 



Sodium chloride solution at 18° C. 



Strength 


Kesistance 


Resistivity 


Conductivity 


Molecular 
Conductivity 


4 normal 


840 ohms 


4-94 


•202 


50-5 


2 „ 


1290 „ 


7-59 


•132 


66 


normal 


2235 „ 


13-1 


•076 


76 


\ normal 


4230 „ 


24-9 


•040 


80 


"k " 


8060 „ 


47-4 


•021 


84 



Draw a curve shewing the relation of conductivity to 
strength. 

By placing the tube in a water-bath, the influence of tem- 
perature on the conductivity of an electrolyte may be found, 
and expressed by a curve with temperatures as abscissae and 
conductivities as ordinates. 



SECTION LXVI. 

CONSTRUCTION OF A CLARK STANDARD CELL, 

Apparatus required: H tube and stand, clean mercury, 
mercurous sulphate, pure zinc, zinc oxide, zinc sulphate, re- 
agents, paraffin and corks. 

In one form of the Clark standard cell the active materials 
are enclosed in a H-shaped tube (Fig. 119), through the lowest 



Cork 
Paraffin 



ZnSO^ cryg 

Hg,SO,r 

Hgr 




I ZDSO4 solution 



ZnSO^ crystals 
ZnHg 



Fig. 119. 

points of which the platinum wires forming the terminals of 
the cell pass. 

The wire on the left (Fig. 119) ends within the tube in a 
small quantity of pure mercury, and that on the right in zinc 
amalgam. The mercury is covered with a paste of mercurous 
sulphate, and this again with a few crystals of zinc sulphate. 
The amalgam is also covered with cr^'stals, and the rest of the 
tube tilled with a saturated solution of pure zinc sulphate. 
Both tubes are closed by cemented corks. 

The following instructions should be carefully attended to. 



320 MAGNETISM AND ELECTRICITY LXVI 

1. To secure purity of the mercury it should be first 
shaken up in a bottle with dilute hydrochloric acid and then 
distilled in vacuo. 

2. To prepare the zinc amalgam add 4 grams of '* com- 
mercially pure " zinc to 36 grams of the mercury, and heat to 
100'' C. in an evaporating dish on a water-bath. If the surface 
of the zinc is clean it quickly becomes amalgamated, and the 
zinc slowly dissolves in the mercury, which should be occasionally 
stirred. If the amalgamation does not occur immediately, the 
zinc should be removed, treated with dilute hydrochloric acid, 
dried and replaced. The amalgam should be liquid at 100° C. 
and solid at ordinary temperatures. 

3. Mix in a flask 40 grams of distilled water with nearly 
twice its weight of crystals of '* pure recrystallised " zinc 
sulphate, and add about two per cent, of zinc oxide to 
neutralize any free acid. Heat gently, never allowing the 
temperature to exceed 30° C. When the crystals have dissolved 
add about 10 grams of mercurous sulphate treated as described 
in 4. Filter the solution while still warm into a stock bottle. 
Crystals should form as it cools. 

The object to be attained is the preparation of a neutral 
solution of pure zinc sulphate saturated with ZnSOj + 7iifi. 
At temperatures above 30° C. the zinc sulphate may crystallise 
out in another form ; to avoid this 30° C. should be the utmost 
limit of temperature. At this temperature water dissolves 
about 1'9 times its weight of the crystals. If any crystals 
remain undissolved they are removed by the filtration. The 
amount of zinc oxide required depends on the acidity of the 
solution, but 2 per cent, will be ample in all cases likely to 
arise in practice with reasonably good zinc sulphate. Another 
method would be to add the zinc oxide gradually until the 
solution became slightly milky. The solution, when put into 
the cell, should not contain any free zinc oxide; if it does, 
zinc sulphate and mercurous oxide are formed on introducing 
the mercurous sulphate, and the oxide may deposit on the 
zinc, and affect the electromotive force of the cell. The 
difficulty is avoided by adding, as described, the 10 grams 
of mercurous sulphate before filtration; this is more than 



LXVI CONSTRUCTION OF A CLARK STANDARD CELL 321 

sufficient to combine with the whole of the zinc oxide origi- 
nally put in, if it remained free. The mercurous oxide formed, 
and any undissolved mercurous sulphate, are removed by 
filtration. 

4. Take about 20 grams of mercurous sulphate purchased 
as pure, and wash it thoroughly with cold distilled water by 
agitation in a bottle ; drain off the water and repeat the process 
at least twice. After the last washing, drain off as much of 
the water as possible. Mix the washed mercurous sulphate 
with a little zinc sulphate solution, and about half its weight 
of pure mercury, adding sufficient crystals of zinc sulphate from 
the stock bottle to ensure saturation. Shake these well up 
together to form a paste of the consistency of cream. Heat 
the paste for an hour to a temperature not exceeding 30° C, 
agitating it from time to time, then allow it to cool, shaking 
it occasionally while it is cooling. Crystals of zinc sulphate 
should then be distinctly visible, distributed throughout the 
mass; if this is not the case, add more crystals from the 
stock bottle, and repeat the whole process. This ensures the 
formation of a saturated solution of zinc and mercurous sul- 
phates in water. 

The above treatment of the mercurous sulphate has for 
its object the removal of any mercuric sulphate, which may 
be present as an impurity. Mercuric sulphate decomposes in 
the presence of water into an acid and a basic sulphate. The 
latter is a yellow substance — turpeth mineral — practically 
insoluble in water; its presence, at any rate in moderate 
quantities, has no evil effect. If, however, it is formed, the 
acid sulphate is also formed. This is soluble in water and the 
acid produced affects the electromotive force of the cell. The 
object of the washings is to dissolve and remove the acid 
sulphate, and for this purpose the three washings described 
will in nearly all cases suffice. If, however, a great deal of 
turpeth mineral is formed, it shews that there is a great deal 
of the acid sulphate present, and it will then be wiser to obtain 
a fresh sample of mercurous sulphate rather than to try by 
repeated washings to get rid of all the acid. The free mercury 
helps in the process of decomposing the acid salt, forming 
8. p. 21 



322 MAGNETISM AND ELECTRICITY LXVI 

mercurous sulphate and sulphuric acid, which will be washed 
away. 

The materials having been prepared, pure mercury and 
zinc amalgam are respectively poured into the two vertical 
parts of the H tube till the platinum wires are covered. The 
amalgam, which is solid at ordinary temperatures, should be 
heated till it is liquid, and the limb of the H tube intended' 
to contain it, heated to about the same temperature. It 
should then be poured into the H tube down a hot glass 
tube of outside diameter less than the inside diameter of the 
H tube, to prevent it soiling the sides of the H tube. The 
mercurous sulphate paste should then be forced down a glass 
tube on to the mercury, and the tube withdrawn, care being 
taken not to soil the H tube. A few crystals of zinc sulphate 
should then be placed on the surfaces of the paste and 
amalgam, and the rest of the tube up to about 1'5 cms. 
from the top filled with the concentrated zinc sulphate solution. 

A small quantity of clean paraffin wax should then be 
melted, and poured gently on to the surfaces of the solution 
in the tubes, till a layer about half a centimetre thick is 
formed. On the top of these layers, corks about '5 cm. thick 
should be placed, and the tops of the tubes then sealed with 
some hot resinous cement. 

A label bearing the date and the name of the maker 
should be attached to the stand on which the tube is sup- 
ported. 

The cell should stand a few days and then be compared 
with a standard cell, and the result recorded on the label. 

A disadvantage of the form of Clark cell which has been 
described, consists in the invariable cracking of the glass 
in the branch containing the zinc amalgam, at the point 
where the platinum is sealed into the glass. This is due to 
the expansion of the platinum as it slowly alloys with the 
mercury. In the cells recently constructed in the Physical 
Laboratory of the Owens College, the platinum wire has been 
covered electrolytically with nickel, to protect it against direct 
contact with the amalgam, and the results so far obtained 
seem to shew that this treatment gets over the difficulty. 



SECTION LXVIL 

COMPARISON OF ELECTROMOTIVE FORCES. 

Apparatus required : Two similar resistance boxes, high 
resistance, mirror galvanometer, Daniell, Leclanchd^ and Clark 
cells, and connecting wires. 

When the electromotive forces of different cells are to be 
compared together, it is necessary to carry out the comparison 
under similar conditions, and the condition usually adopted is 
that the cell under test shall be giving no current, or only an 
extremely small one, at the time of the test. The electromotive 
force E^ of the cell must therefore be balanced, the balance 
being indicated by a galvanometer in series with the cell 
remaining undeflected. The balancing electromotive force is 
best provided by the difference of potential between two 
points of a resistance through which a current is passing, and 
wires from the cell under test are brought to these two points. 
If it, is the resistance between the points of contact, C the 
current through that resistance, and E the difference of poten- 
tial between the points, we have 

El = CRi. 

If a second cell of electromotive force E^ is substituted, 
and balance exists when the resistance between the points of 
contact is /?,, we have, if the current is the same, 

E, = CR,. 
Hence 

E^ U^ 

which gives the ratio of the electromotive forces. 

21—2 



f 



324 MAGNETISM AND ELECTRICITY LXVII 

Connect two similar resistance boxes A and B (Fig. 120), 
each of about 10,000 ohms, in series 

with a plug key K and two Leclanch^ ^ T I 

cells, taking out the plug from the ^11 '' 

key before making connexions. Set © 
up a Daniell cell, with a clean zinc j^" 

plate in zinc sulphate, and a clean 
copper plate in copper sulphate. To 
the terminals of one of the boxes, say j,. ^20. 

A J connect, in series with each other, 

the Daniell cell, a galvanometer G, a resistance R of about 
100,000 ohms, and a spring key K.^, arranging that the cell 
under test, if alone, would send a current through the box A in 
the same direction as that sent by the Leclanch^s. 

Take out plugs for 10,000 ohms from the box A. Insert 
the plug in the key K, make connexion for an instant at 
the spring key Kc, in the test cell circuit, and observe the 
direction in which the spot of light moves. Take out plugs 
for 5,000 ohms from the box B, plugging 5,000 ohms in A 
so that the sum of the resistances of A and B remains the 
same. Make contact at the key K^, and notice the direction 
of motion of the spot. If it is the same as previously, take 
out more plugs from B and insert the same number in A. 
Continue adjusting the resistance of the two boxes, keeping 
the sum constant, till on closing the galvanometer circuit there 
is no deflection. If the arrangement is not sufficiently sensitive 
to enable the correct resistance to be found to within 1 ohm, 
determine it as nearly as possible, then short circuit the 
100,000 ohms in the galvanometer circuit, and determine it 
more accurately. Make a note of the resistance in each box. 

Insert again the 100,000 ohms in the galvanometer circuit, 
then replace the Daniell by a Leclanch^ cell and balance as 
before. 

Then substitute the Clark cell, reading the tempera- 
ture of the air in the neighbourhood of the Clark cell, 
or better still, placing the cell in a water or oil bath, the 
temperature of which is measured. Since a Clark cell made 
according to the instructions contained in the Memorandum 



LXVII 



COMPARISON OF ELECTROMOTIVE FORCES 



325 



of the Board of Trade (Section LXVI.) has a difference of 
potential at its electrodes of 1-434 — '001 {t — 15°) volts, where 
t is the temperature centigrade, the electromotive force of each 
of the other cells can be found in volts. 

Calculate the electromotive force of the Daniell and of the 
Leclanche cells in volts. 

Arrange observations and results as follows : — 

18 January, 1896. 
Resistance Boxes A and B. 



Cell 


Besistance A 


Resistance B 


Sum 


Daniell, No. 4 
Leclauche, No. 13 
Clark, No. 3, 19° C. 
Leclanche, No. 13 
Daniell, No. 4 


3,248 
4,124 
4,202 
4,122 
3,246 


6,752 
5,876 
5,798 
5,878 
6,754 


10,000 

» 
» 



electromotive force of Clark at 19°= 1430 volt, 
electromotive force of Daniell 



0947 



4123 
„ Leclanche = 1*43 x ^^^ = 140 

The effect of temperature on the Daniell and Leclanche 
is masked by other irregularities, hence the temperature need 
not be noted and an accuracy of one per cent, in the result is 
sufficient. 

A compact fonn of the apparatus used in this exercise is 

known as a "Potentiometer." In it the re- 

sistance A consists of 15 equal resistances in i ± 

series, the one at the B end being a wire ~ 

along which the contact to the galvanometer 

slides. The contact from the cell to be 

tested is made by a switch moving over the 

15 contact pegs at the ends of the 14 equal 

resistances. The arrangement is shewn in a 

diagiamniatic form in Fig. 121. The rough 

adjustment for a balance is made by means of the switch and 

the fine adjustment by means of the slider on the wire. 



^ 



i 



Fig. 121. 



326 



MAGNETISM AND ELECTRICITY 



LXVII 



If the main current is supplied by a battery of two 
Leclanch^s, the above method may be used to measure any 
electromotive force less than about three volts, e.g.^ that at 
the terminals of a voltmeter intended for accumulator cells. 

When however the electromotive force to be measured 
exceeds a few volts, as, e.g., when a voltmeter reading to 
100 volts is to be standardised, the arrangement requires 
modifying slightly because the electromotive force of the 
battery supplying the current cannot be raised above about 
three volts for fear of overheating the resistances A and B. 

The fall of potential between the terminals of the volt- 
meter is subdivided by connecting to the terminals a high 
resistance, divided into a number of parts the resistances of 
which bear known ratios to that of the whole. The difference 
of potential at the ends of one of these parts is then compared 
with that of the Clark cell by the above method. The circuit 
is arranged as shewn in Fig. 122, where V is the voltmeter 



1-^0)—^ 



1 






Fig. 122. 

to be tested, R a variable resistance through which it is 
connected to a battery E, r the high resistance, r^ the portion 
of it down which the fall of potential is measured. If e is the 
value found by experiment, the electromotive force at the ter- 



minals of the voltmeter = — e. 



SECTION LXVIII. 

THE POTENTIOMETER METHOD OF MEASURING 
CURRENTS. 



Apparatus required : Standard low resistance, adjustable 
resistance, storage cells, current measuring instrument to he 
standardised, Clark cell and mirror galvanometer. 

When an electric current of A amperes is sent through a 
resistance of R ohms, it creates a difference of potential of V 
volts between the ends of the resistance, where V=AR, and 
sometimes it is more convenient to measure the current by 
means of the difference of potential it produces at the euds of 
a known resistance, than to measure it directly. This method 
is known as the " potentiometer method." In order that the 
method may give accurate results, the resistance R should be 
made of a material having a small tempemture coefficient, and 
should have a sufficiently large surface to prevent the tem- 
perature rising more than a few degrees. 

We shall shew how the method is used to standardise a 
current metre. 

Ck)nnect the current measuring instrument ^, to be tested, 




Fig. 128. 

to a standard low resistance R, capable of carrying the current 
which it is propo.sed to use without undue heating, and to an 
adjustable resistance /J,, using a number of storage cells E 
to supply the current required (Fig. 123). 



328 



MAGNETISM AND ELECTRICITY 



LXVIII 



Connect the terminals of a voltmeter which has been 
standardised (Section LXVII.) to the ends of the standard 
resistance. If such an instrument is not available, or if it is 
necessary to carry out the estimation of current to a high degree 
of accuracy, connect the ends of the resistance through a mirror 
galvanometer and tapping key to the ends of one of two similar 
resistance boxes A' , B' arranged in series, and forming a circuit 
with two Leclanche cells. Find, as described on p. 324, the 
plugs which must be taken out of the two boxes to enable 
contact at the key to be made, without a deflection of the 
galvanometer resulting. Now substitute in the galvanometer 
circuit a Clark cell e for low resistance R, and again balance 
by adjusting the resistances in the two boxes keeping their 
sum constant. If a reliable "Potentiometer" is available, it 
may be used instead of the two resistance boxes to compare the 
electromotive forces as explained on page 325. 

Observe the temperature of the Clark cell, calculate its 
electromotive force at that temperature. From the two obser- 
vations of resistances in the boxes A and B calculate the 
electromotive force E at the ends of the standard resistance. 
If the resistance of the standard is R ohms, the current 
through the resistance and current measuring instrument is 
E/R amperes, and this should be compared with the current 
as registered by the instrument. 

Record as follows : 

16 January, 1897. 

Standardisation of Ammeter No. 4. 

Standard low resistance C = '20 ohms. 

Clark cell No. 4 at 18° = 1-431 volt. 





Resist- 


Resist- 






Reading 


Cor- 


Connexion 
to 


ance m 
A' 


ance in 


E.M.F. 


Current 


on Am- 
meter 


rection 


Clark cell 


6560 


4551 


1-431 








Ammeter 


956 


10155 


•207 


1-03 


1-01 


-h-02 


)j 


1940 
ifec. 


9161 


•418 


2-09 


205 


+ •04: 


Clark cell 


6559 


4552 


1-431 









SECTION LXIX. 

THERMO-ELECTRIC CIRCUITS. 

Apparatus required : Watei- baths, thermo-circuits, four- 
way key and tnh'ror f/alvanomet€7\ 

If a circuit consists of wires of different materials, and if one 
of the junctions of two dissimilar wires is heated, an electric 
current flows through the circuit, and continues to flow so long 
as the difference of temperature between the heated junction 
and the rest of the circuit is maintained. 

This electric current is due to an electromotive force 
produced by the inequality of temperature, and it is found, 
for small differences of temperature between the two junctions, 
to be nearly proportional to the difference. For gi-eater differ- 
ences, if ti is the temperature of the hot junction, t^ that of the 
rest of the circuit, the electromotive force e in a circuit of two 
metals is given by the equation 



= ^(«.-o(?'-''-|^") 



where A and T are constants depending on the two materials 
of the circuit, T being a temperature known as their " neutral 
temperature." 

To verify the above statements the apparatus shewn 
(Fig. 124) is provided. 

It consists of two vessels containing water, in which are 
placed two test tubes containing the junction of the wires to 
be experimented on, and thermometers for indicating their 
temperatures. The rest of each tube is filled with clean sand 



330 



MAGNETISM AND ELECTRICITY 



LXIX 



or with petroleum, to improve the thermal connexion of the 
junctions and thermometers with the water. 




Fig. 124. 

The circuits to be tested consist of lengths of iron, nickel, 
and platinoid wire, to the ends of which lengths of copper 
wire are soldered, and brought to binding screws placed on the 
board through which the test tubes pass. 

The binding screws should be joined by copper connecting 
wires to a four-way key, so that each circuit may in turn be 
connected to a galvanometer of about 50 ohms resistance. 
With a galvanometer of this resistance, the effect of the 
different resistances of the circuits may be neglected, and the 
deflections taken as proportional to the electromotive forces 
acting in the various circuits. 

Fill the two vessels with water at the temperature of the 
room, and connect the thermo-circuits in turn through the 
four-way key to the galvanometer. Verify that there is no 
current in any of the circuits. 

Now raise the temperature of the vessel which has no 
binding screws over it, about 10°C. and keep it constant for 10 
minutes. Then connect the galvanometer to each circuit in 
turn, and determine the deflection, noting the temperature 
before and after each observation. Raise the temperature 
10° C. further and repeat. 



LXIX 



THERMO-ELECTRIC CIRCUITS 



331 



Continue till the hotter vessel reaches about 70" C, then 
cool it by adding cold water, taking observations in the same 
way during the process. 

Record as follows : 

29 January, 1889. 



'o 


'i 


# 


Deflections 




Copper iron 


Copper nickel 


Copper platinoid 


17-0 


28-5 


-1-5 


+ 3-3 


+ 2-6 


170 


39-5 


-30 


6-3 


50 


171 


52-0 


-4-7 


11-4 


9-2 


171 


63-0 


-6-2 


14-9 


12-4 


171 


700 


-7-2 


18-0 


14-3 


171 


57-0 


-5-5 


12-6 


10-2 


171 


49-0 


- 4-5 &c. 


9-7 ifec. 


7-8 Ac. 



Represent the observations for each circuit by a curve 
taking tj — U as abscissae and the deflections as ordinates. 
The curves will be found to be almost straight lines, and 
from this it is seen, that if we put the equation (p. 329) into 
the form 

A 



e = J5:o(f,-fo)(l-t^i + 0,where^o = -4^and6 = 



2E, 



b must be small. 

Assuming that 6 = 0, determine the values of Eq in galvano- 
meter scale divisions per degree, which most correctly reproduce 
the curves found experimentally, and taking its value for the 
copper wire circuit to be unit}^ find its values for the other 
two circuits. 

If more accurate results are required, a low resistance 
galvanometer must be used, and the resistance of each circuit 
be made equal, or the total resistance of the galvanometer and 
each circuit be taken into account in comparing the electro- 
motive forces of the circuits. 



SECTION LXX. 

MEASUREMENT OF THE MECHANICAL EQUIVALENT 
OF HEAT BY THE ELECTRICAL METHOD. 

Apparatus required :, Covered calorimeter, thermometers, 
heating coil, standardised ammeter and voltmeter, storage cells, 
watch. 

When the whole of the work done on a body is converted 
into heat, the amount of work done bears a fixed ratio to the 
amount of heat produced, in whatever way the work is performed, 
and the work which has to be done to generate one gram-degree 
of heat is, we have seen (p. 142), known as the "mechanical 
equivalent " of heat. To determine this quantity, any convenient 
method of generating heat by performing work on the body may 
be adopted, and it is proposed in this section to do work by 
sending a current of electricity through an insulated wire 
immersed in water. 

If A is the current passing through the wire, and E the 
electromotive force at the ends of the wire, the rate at which 
work is done on the wire per second is EC watts, and if 
the current flows for t seconds, the total work done = ECt 
joules = ECt . 10"^ ergs. If the water rises in temperature 
degrees, and the water equivalent of the calorimeter thermo- 
meter and coi\ = w, the heat generated, supposing no heat is 
lost by radiation etc. = wO. If / joules are necessary to 
generate one gram-degree of heat, we have 

EAt = wdJ, 
from which J can be found. 

Weigh the calorimeter and stirrer provided. Nearly fill the 



LXX MECHANICAL EQUIVALENT OF HEAT. 

calorimeter with water and weigh again. Weigh also the 
platinoid resistance coil, and support it from the wooden lid 
of the calorimeter, taking care that it does not touch the sides. 
Place a thermometer graduated in tenths of degrees in the 
water. Connect the coil through a standardised ammeter and 
an adjustable resistance to sufficient storage cells to furnish the 
current required. Connect a standardised voltmeter of known 
resistance to the ends of the resistance coil. Make circuit and 
see that the instruments give proper indications and that the 
thermometer shews a gradual rise of temperature. Break the 
circuit, stir the water well, and after a few minutes take obser- 
vations of temperature every half minute as described on p. 129. 
At the end of the first period put on the current, read the 
thermometer every half minute, observing the voltmeter 15 
seconds before each minute and the ammeter 15 seconds after 
each minute. This second period should continue till the tem- 
perature of the calorimeter has risen about three degrees, then 
at the end of one of the intervals, the current should be switched 
off and observation of temperature continued till the rate of 
change is steady. From the first and third periods the cooling 
corrections during the second and third periods should be 
calculated as in pp. 129 — 130, and from the initial and final 
corrected temperatures the rise of temperature determined. 

J is calculated by substituting in the above equation for E 
the mean electromotive force, and for A the mean current, if 
both quantities show only small variations during the ex- 
periment. If they are not sufficiently constant, the product of 
EA must be calculated for each interval of time and the mean 
product substituted in the equation. . 

Record as follows: 

10 December, 1897. 

Weight of calorimeter = 550 grams 

„ „ and water ... =245*2 „ 

water =1902 

„ platinoid coil ... ... = 28 „ 

Water equivalent of calorimotcr find 

coil . = 7-8 



334 



MAGNETISM AND ELECTRICITY 



LXX 



Total water equivalent 

Rise of temperature . . . 

.*. Heat generated ... 

Mean electromotive force 

Resistance of voltmeter 

Mean current through ammeter 

„ „ „ voltmeter 

coil ... 



1-45 
14 



= 198 grams 

= 311° C. 

= 616 gram-degrees 

= 1'45 volts 

= 14 ohms 

= 11 10 amperes 

= 10 „ 



... = 1100 
Time ... ... ... ... ... =161 seconds 

Work done = 1'45 x 11 x 161 ... = 2575 joules 

. •. Equivalent = ^-^ = 418 joules per gram-degree 

= 41'8 X 10^ ergs per gram-degree 

This value happens to be almost exactly right, but errors of 
one per cent, are likely to occur, unless the voltmeter and 
ammeter have been very carefully standardised. 



SECTION LXXI. 

INDUCTION OF ELECTRIC CURRENTS. 

Apparatus required : Two solenoids, one sliding within the 
other y tangent galvanometer, reversing key, low resistance mirror 
galvanometer, resistance coils. 

When a current is made, broken, or altered in strength in 
any circuit, induced currents are produced in neighbouring 
circuits, and it is the object of this exercise to find on what 
conditions the magnitudes of these induced currents depend. 

The induced currents will last only a very short time, and 
a galvanometer in one of these neighbouring circuits will not 
indicate the strength of the current, but the total quantity of 
electricity which has passed through it. If a is the angle of 
the first swing of the galvanometer needle produced, it can be 
shewn that the quantity of electricity which has passed through 

the coil of the galvanometer is proportional to sin ^ . If the 

galvanometer needle hangs in its proper position when no 
current passes through the instrument, and the scale is properly 
adjusted, 

where x is the observed deflection and d the distance of the 
scale from the mirror (page 160). If an extreme error of a 
half per cent, is considered allowable, and the deflections ob- 
served on a scale half a metre in length, placed a metre from 
the mirror, never exceed 25 cms., the second term on the right- 
hand side is negligible ; and we may therefore take the observed 



336 



MAGNETISM AND ELECTRICITY 



LXXI 



reading x to be proportional to the quantity of electricity 
which has passed through the galvanometer. The experiments 
of the present exercise are supposed to be made under these 
conditions. 

Two solenoids, P, S (Fig. 125), mounted on blocks of wood 
so that their axes are coincident, are provided. The inner coil 




Fig. 125. 

P can be moved in a direction parallel to the axis, and placed 
with its centre at any convenient distance from the centre of 
the outer fixed solenoid >S^. Each solenoid is divided into three 
parts, and the number of turns to each part should be counted 
and recorded. 

Arrange the turns of the inner coil in series with each other 
and with a cell, a reversing key, an adjustable resistance, and a 
tangent galvanometer. Place it within the outer coil so that 
the centres of the coils coincide, the mark on its base will 
then read on the scale of the outer coil. Connect, by 
means of fairly stout copper wire, the end terminals of the 
outer coil through a resistance box to a low resistance mirror 
galvanometer. At first cut the mirror galvanometer out of 
circuit by connecting the two wires leading to it to the same 
terminal, and observe whether making or breaking the battery 
circuit has any effect on the galvanometer. If so remove the 
coils further away and put their axis in such a position that 
this is no longer the case. 

Adjust the resistance in series with the inner coil till the 
current flowing through it is one ampere, and take readings 
occasionally to see that it remains steady. Notice that making 
or breaking the battery circuit produces a deflection of the mirror 
galvanometer needle. Observe the extent of the first swing and 



LXXI INDUCTION OF ELECTRIC CURRENTS 337 

verify that the swing on making is equal and opposite to that 
on breaking the primary, t.e. the battery circuit, and that the 
swing on reversing the primary current is double the previous 
swings. If this is found not to be the case, cut the galva- 
nometer out of circuit as previously and make sure that there 
is no direct action from the primary coil. 

Effect of the relative positions of the two coils. Arrange 
the resistances so that on making or breaking one ampere in 
the primary, the deflection is about half the greatest observable 
deflection. Determine its amount, then slide the inner coil 
through 1 cm. in a direction parallel to the axis of the coils, 
and again determine the deflection. Repeat for 2, 5, 10, 15 
and 20 cms., thus gradually sliding the inner coil out of the 
outer one. 

Now rotate the movable coil till its axis is bisected at right 
angles by that of the fixed coil, and verify that there is no 
deflection on making or breaking the primary circuit. 

Move the inner coil towards the outer, keeping it parallel 
to itself, and verify that by proper adjustment a direction of 
the axis may be found, for which there is no doflection, even 
when the coils are near together. 

Effect of the resistance of the secondary circuit. Place the 
movable coil within the fixed coil in the position in which the 
induced current has been found to be a maximum, and deter- 
mine the swing on making or breaking the primary circuit. 

Double the total resistance in the mirror galvanometer (or 
secondary) circuit by taking out plugs from the box equal to 
the sum of the resistances of the galvanometer, coil and con- 
necting wires, and verify that the swing is now half what it 
was before. 

Effect of the magnitude of tfie current in the pj-iniary. 
Increase the resistance in the battery circuit till the current 
has half its original value, and verify that on making or 
breaking, the swing is half what it previously was. 

Effect of number of turns in the primary. Adjust the resist- 
ance till the primary current is again one ampere. Change the 
connexions of the primary so that the current passes through 
tiitro coils only. The connexions are arranged so that the 
s. p. 22 



338 MAGNETISM AND ELECTRICITY LXXI 

resistance in cii-cuit, and therefore the current, remains the 
same; but if any variation is observed, adjust the resistance 
in series with the storage cell till the current is again one 
ampere, and observe the swing on making or breaking. Notice 
that it is decreased in the ratio 3 : 2. Now connect to the two 
centre screws so as to send the current through the central coil 
only. Notice whether the current remains the same ; if so the 
deflections will be found to be again decreased in the ratio 2:1. 

Effect of number of turns in the secondary. Returning to 
the whole of the turns on the primary, change the connexion 
of the galvanometer from 3 to 2 coils of the secondary. The 
terminals of the coils are so arranged that the resistance in 
circuit remains the same. Verify that on now making or 
breaking the primary circuit, the swing is decreased in the 
ratio 3 : 2. Make connexions to the centre coil only, the ar- 
rangement being again so that the resistance remains constant. 
The deflection will be found to be decreased in the ratio 2 : 1. 

It has therefore been shewn that for a given position of 
the coils with respect to each other, the induced current in 
the secondary is proportional to the current in the primary, 
to the number of turns of the primary, to the number of turns 
of the secondary, and inversely proportional to the resistance 
of the secondary circuit. 

Effect of moving magnets. Instructive experiments maybe 
made, when two observers are available, by moving magnets 
in the neighbourhood of the secondary coil. For this purpose 
the coil should be removed to a distance, such that the bar 
magnet, to be used in the experiments, has no direct effect on 
the galvanometer. 

Place the magnet in the centre of the coil, let one observer 
remove it quickly to a distance from the coil, and let the other 
observer note the deflection produced. 

Replace the magnet in its original position and withdraw 
it again, but pulling it out towards the other side. This should 
produce an equal deflection and in the same direction. 

Place the magnet outside the coil and in such a position 
that its axis coincides with that of the coil ; note the deflection 
when it is suddenly moved to a distance, and shew that it is 



LXXI INDUCTION OF ELECTRIC CURRENTS 339 

the same as if the magnet were turned through a right angle 
about its centre. 

Verify that when a magnet is moved quickly from a position 
A to a. position B the effect on the galvanometer is the same» 
through whatever path the magnet is moved. Also try to 
shew that when the magnet can be moved from a position A 
along any path back to the same position -A in a time which is 
short, compared to the time of oscillation of the galvanometer 
needle, the total effect on the latter is nil. 



22—2 



SECTION LXXIL 

DETERMINATION OF THE INDUCTANCE OF A COIL. 

Apparatus required : Coil of considerable self-induction. 
Post Office resistance box, additional resistance, mirror galvano- 
meter, voltaic cell of constant electromotive force. 

If one of the arms of a Wheatstone bridge possesses self- 
induction, the galvanometer will, even when the bridge is 
balanced for steady currents, shew a temporary deflection, when 
the strength of the current through is altered. 

If L is the inductance of the coil and x the rate of change 
of the current passing through it, the effect on the galvanometer 
is equal to that which would be produced if an opposing electro- 
motive force Lx were introduced into the arm in which the 
coil is placed. Such an electromotive force would cause a 
current KLx to flow through the galvanometer, where the value 
K depends on the resistances of the galvanometer, and of the 
arms of the bridge. If the current is suddenly changed from 
to X, the electromotive force at each instant depends on the 
rate of change, but the total quantity of electricity, which flows 
through the galvanometer, will be KLx, which is the same as 
that conveyed by a constant current KLx, flowing during a 
short time t, such that xt = x. 

This quantity of electricity passing through the galvano- 
meter coils, will cause a deflection or " throw " of the galvano- 
meter needle, the extent of which may be calculated. 

If the moment of inertia of the suspended system of 
magnets be /, and (o denote the initial velocity, ^Ico^ is the 
initial kinetic energy. The diminution of kinetic energy during 
the swing must be equal to the work done against the magnetic 



LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 341 

forces tending to bring the system back into its position of 
equilibrium. To find an expression for that work, we may 
think of the analogous case of a compound pendulum. If the 
weight of such a pendulum be W, and the difiference in level of 
the centre of gravity between the displaced and equilibrium 
position 6, the work done will be Wb, which is equal to 
Wh ( I — cos 6), where 6 is the angular displacement, and h the 
distance of the centre of gravity from the point of suspension. 
Hence for the greatest angular displacement (a), for which the 
kinetic energy is zero, the work done must equal the original 
kinetic energy, or 

/&)- = 2ir/t (1 - cos a) = 4Tr^sin» |, 

.-. a, = 2y -p.sin^. 

Returning to the case of the oscillating system of magnets, 
we need only replace Wh by the moment of the couple, tending 
to bring the magnets back, when the displacement is a right 
angle. Representing this quantity (which is equal to the 
product of the earth's horizontal force and the magnetic 
moment of the system) by H, we obtain for the relation be- 
tween the initial angular velocity and the greatest angle of 
displacement, 

a 



.-V? 



sin^ 



If G denote the couple acting on the galvanometer needle 
in its position of rest, when unit current pivsses through its 
coils, a current i will cause an angular acceleration Cri/I, and 
if the duration of this current be t, which is supposed to be 
small compared to the time of oscillation T of the galvanometer 
needle, the angular velocity generated will be Oir/I, where 
we may replace it by the total quantity of electricity Q, which 
has passed through the galvanometer coils. Substituting this 
angular velocity in the above equations we find 



342 MAGNETISM AND ELECTRICITY LXXII 

The moment of inertia / may be determined or eliminated, 
by observing T, for 

Hence 

The value of HIG can be determined experimentally by 
observing the permanent deflection 6 of the galvanometer, 
produced by a known current z, for in that case (p. 243) 

Gi^Htaxie, 

provided that the galvanometer needle moves in a field which 
is sufficiently uniform for the tangent law to be true (see 
page 244). Otherwise tan 6 must be replaced by some function 
of 6, obtained by calibrating the instrument. 

If the simpler supposition is sufficiently accurate, 

IT tan 6 

We have deduced the value of a under the condition that 
the needle moves without damping, which is never quite the 
case. But the damping may be taken into account by multi- 
plying the observed amplitude by e^'"^, since the amplitude of 
an oscillating system is reduced in the time t in the ratio of 
e-^tiT.i^ p. 282, and the time taken up by the needle, in 
moving from its position of rest to its first greatest elongation, 
isr/4. 

The final equation for Q is therefore 

^zV/2siu| 
TT tan 6 

To apply this equation to the determination of the inductance 
of a coil we substitute for Q the value KLx, which has been 
shewn to be equal to the quantity of electricity flowing through 
the galvanometer, when the current in the branch of the Wheat- 
stone bridge containing the self-induction L is suddenly changed 



LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 343 

from to X. The quantities K and % are eliminated in the 
following way. If the balance of the bridge be disturbed, by 
inserting a small resistance r in series with the coil, this addi- 
tional resistance will, while a current x' is passing through it, 
have the same eftect on the galvanometer as an opposing 
electromotive force rx. If this electromotive force produces 
a current i in the galvanometer, then i = Krx'. Hence 

L = r — , ^ . 

X TT tan V 

This equation holds within the limits of deflection for which 
the tangent law is true. 

In many cases the calculation may be further simplified. 
The ratio x'lx is the ratio of the currents in the coil with and 
without the additional resistance r ; the latter quantity should 
always be so small that x is sensibly equal to x. 

If a and Q are sufficiently small these angles may be taken 
to be proportional to the observed deflections d^ and 2), otherwise 

sin ^ and tan 6 must be evaluated according to the equations 

given on p. 160. But it follows from these equations that the 
ratio sin a/2 / tan B is equal to c?i/2(/o to the second approxi- 
mation if 



■|V¥=— 



This may be secured by a proper choice of r. 

The coefficient e^^^* may be expressed in the form of a series, 
the first three terms of which are 

^ + 2+8- 

The third term is often negligible, e.g. when an accuracy of !•/, 
is considered sufficient and X is smaller than '2, or when an 
accuracy of P/o is required and \ is smaller than 06. 

The determination of X is best carried out by a separate 
series of observations, but when that quantity is small, and 



344 MAGNETISM AND ELECTRICITY LXXII 

when the experiment is made rather as an exercise than for 
the purpose of obtaining the best possible results, time may 
be saved by observing not only the first throw (d^), on making 
or breaking circuit, but also the second deflection (d.^) on the 
same side. The equations on p. 282 give for this case d^/d^ = 6^^ 

and hence e^/*^ may be replaced by 1--^] . When di and d^ are 

nearly equal we may simplify further, for 



r =(■*''■**)' 



- 1 4. ^ d,-d, 5^ ( d, - cg, y 

~ "^4 d, "^32 1 d, ; "^••• 

If the third term of the series can be neglected, we obtain 

d,e'l^ = d,-\-l{d,-d,). 

If an accuracy of one per cent, is required, this simplification 
should not be introduced, unless d^jd^ is smaller than \'2 which 
corresponds approximately to X smaller than "1. 

Taking account of these simplifications we shall be able 
to use one of the following three equations whenever X is 
smaller than '2, i.e. d^jd^ smaller than 1*5, and d^jd^ is approxi- 
mately equal to 1"17. 

r_Trd, (d,\i 
27rI»'W ' 

^-^- D • 

The first equation is the one to apply, when accuracy is of 
greater importance than time, so that an independent de- 
termination of X may be made. The second is used when X is 
not separately determined, but when the second deflection is 
observed ; while the third applies to the same case with the 
additional proviso, that djdz is smaller than 1*2. 

Care should be taken, to have the zero reading of the 
galvanometer near the centre of the scale and to take the 



LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 345 

deflections towards both sides, thus eliminating any errors due 
to faulty adjustment of the scale. 

If X is to be determined by experiment, we make use of the 
equation (p. 283) 

log s, - log Sn = (n-1)\ log e, 

8i and Sn being the first and the nth arc of swing. It is found 
that an error of reading has the smallest effect when 

5,/5„ = e=2-7. 

This suggests the following procedure. Set the galvano- 
meter needle into vibration, and read the first deflection (i, on 
the right and the first deflection rf/ on the left. Then wait until 

the deflection has diminished to about ^ and read again 

deflections dn and rf,/. Assuming as a first and generally suffi- 
cient approximation, di -|- d^ and rf„ + dn, to be proportional to 
the arcs of swing, we may, knowing n, apply the above equation 
to the calculation of \. The result may be rendered more 
accurate by observing and combining in a similar manner 5, 
and «„+!, «s and s^+j etc. If the oscillations are too rapid to 
allow the deflections toward both sides to be accurately ob- 
served, we must divide the experiment into two parts, observing 
first deflections to the right and applying the equation 

log rfi — log rf„ = 2 (/I — 1 ) \ log e 

to deduce the value of \. Observing similarly the deflections to 
left we obtain a second value for X, the mean of the two being 
taken for the final result. When \ is so sfnall, that it would 
take too long to wait till the amplitudes are reduced in the 
above ratio, we may procecnl as in the measurement of time 
intervals, observing say six deflections, waiting until six oscilla- 
tions have passed unob.served, and observing six further ones, 
which are combined with the six first in the manner already 
explained. 

The principal experimental difficulty will be found to be 
due to the gradual heating of the coil by the current, which 
must be sufficiently strong to give a measurable throw on 



346 MAGNETISM AND ELECTRICITV LXXII 

making or breaking the circuit. Subject to this condition, the 
current through the coil should be as weak as possible. It is 
important therefore that the arms of the bridge should be 
arranged, so that the galvanometer is most sensitive to variations 
of resistance in one arm of the bridge, for a given current in 
that arm. This condition leads to results, which are quite 
different from those which apply when the electromotive force 
is given (p. 287). The best arrangement of the bridge, when S 
is the resistance whose inductance is to be determined, is that 
in which the resistance we have denoted by Q in the previous 
exercises is as small as possible, and that denoted by R as 
large as possible \ One of the galvanometer terminals should 
be connected to the junction of P and Q, the other to the 
junction of R and 8. The best resistance of the galvano- 
meter of a given type is 



G^ = 



p+e 



or when P is large compared to Q, G = S -\- Q, S being the 
resistance of the coil. The sensitiveness of the arrangement 
for given values of P, Q and G, as compared to the best possible 
arrangement, is given by the fraction 




where k is the ratio of the actual to the best possible galvano- 
meter resistance. A discussion of this equation shews that the 
sensibility is not much reduced, if k is intermediate between 4 
and i', also that it will be sufficient to make the ratio PIQ== 10. 
But care should be taken that Q never exceeds S and it would 
be better if S were at least three times as great as Q. 

The change of resistance r may be obtained by connecting 
a high resistance in parallel with the coil, when the steady 
deflection is taken. If W be the resistance of the shunt, 
r is equal to S'''/{S -}- W). The shunt must be disconnected 

1 Arthur Schuster, "Electrical Notes," Phil. Mag. p. 175 (1894). 



LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 347 

when the throw is taken. In order to avoid a change of 
connexions during the observations we may vary the resistance 
of R instead of that of S, remembering that a change r in the 
resistance of the coil is equivalent to a change PrjQ in the 
resistance R. If r' is the change in R, which produces a steady 
deflection 0, we must therefore substitute for r the quantity 
r'Q/P. 

To carry out the experiment, arrange the given coil S to 
form the fourth arm of the Post Office bridge; Q being the 
ratio arm adjoining S. Make Q=10 and P=1000. Connect 
the galvanometer through a commutator to the junction be- 
tween P and Q, and to that between R and S. Introduce also 
a commutator into the battery circuit, so that the current may 
easily be reversed. It is important that the battery should be 
one giving a constant electromotive force, e.g. a Daniell or 
storage cell. Leclanche cells are too variable. Adjust the zero 
of the galvanometer to the centre of the scale nearly. 

Leave the galvanometer circuit open, pass the current 
through the coil and notice whether the galvanometer shews 
a deflection. If it does, the coil acts directly on the needle 
and its position should be changed so that the action ceases. 
This is most easily secured by placing the coil with its plane 
nearly horizontal and tilting it till the magnetic field due to it 
is vertical at the galvanometer. Now close the galvanometer 
circuit and balance for steady currents. 

When the balance is approximately made, break the battery 
circuit and take an observation of the throw, which should be 
between 10 and 20 cms. as measured on the scale. If the 
throw is too large, the electromotive force of the battery should 
be diminished or a resistance inserted in the battery bninch. 
If it is too small, the galvanometer should be made more 
.«iensitive, and if that is not possible the electromotive force of 
the battery must be increased. Having obtained an approximate 
value of the throw, the change of resistance r' in the arm iJ, 
which gives a steady deflection of between I'l and 1*2 times 
that of the throw, should be found. 

If the galvanometer has a small logarithmic decrement and 
takes a long time to come to rest a small coil in series with an 



348 MAGNETISM AND ELECTRICITY LXXII 

auxiliary battery and key may be fixed behind it, and con- 
nexion may be made and broken in such a way as to bring the 
suspended system to rest. 

The above preliminary experiments having been concluded, 
note the position of rest of the galvanometer needle. Obtain 
as accurate a balance of the bridge as possible. If there is a 
slow creeping of the spot of light, while the steady current 
passes through the coil, it is probably due to the heating action 
of the current, which slowly increases its resistance. If that is 
the case, adjust the balance so that the spot of light is deflected 
a little to the side opposite to that towards which the creeping 
takes place. Wait till owing to the heating action the spot of 
light occupies accurately the zero position; then suddenly break 
the battery circuit and observe the throw, reading both the first 
deflection (c^) and that which follows it on the same side (d^). 
Make the battery circuit again, change the resistance R by 
the addition of r' and observe the steady deflection. Break 
the galvanometer and battery circuits and read the zero 
again. 

Reverse the commutator in the battery circuit and repeat 
the observations, both for the throw and the steady deflection. 

Reverse the galvanometer current and again repeat the 
observations, first with the battery commutator in one and 
then in the other position. 

Four sets of readings are thus obtained the means of which 
must be combined in the final result. 

Determine the time of oscillation of the galvanometer needle 
by observing the time of say 50 swings if the time be short and 
the logarithmic decrement small ; or if the time be long, observe 
the instant of passage through zero for say six swings, allowing 
an interval of six, then observe six more and combine in the 
usual way. 

If it is desired to determine the logarithmic decrement this 
should now be done by one of the methods explained above, 
care being taken that the sensitiveness of the galvanometer is 
the same as that used during the experiment. The logarithmic 
decrement for the same galvanoineter is proportional to its time 
of oscillation. 



LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 349 

Tabulate your results as follows : 

2 January, 1898. 

Inductance of coil A. Galvanometer No. 327. 

P = 1000. Q = 10. 2 Daniell Cells. 

Balance obtained with i2 = 2194. 

On breaking circuit 

First swing to right (mean of four observations) = 11*04 cms. 
Second „ „ „ „ „ = 7*56 „ 

Changed R to 2255 ; r = 61, .-.?'= 61. 

Steady Deflection (mean of four observations) =12*11 „ 
Time of oscillation = 3*44 sees. 

J _ n. 3*44 1104 /ll*04\i 
• ^-'^^ 27r-i2lll75¥; 
= 335 henries. 



SECTION LXXIII. 

LEAKAGE AND ABSORPTION IN CONDENSERS. 

Apparatus required : Condensers, one of which is specially 
selected to shew leakage and absorption, high resistance galva- 
nometer, discharge key, cells. 

Unless the plates and terminals of a condenser are well 
insulated, the condenser is found gradually to lose its charge. 
The rate at which this loss takes place is generally greater, the 
greater the charge which the condenser possesses. 

To determine the extent to which leakage takes place in 
a condenser C (Fig. ]27), connect it 
through a " condenser key" K to one or 
two Leclanche cells L and a galvano- 
meter G, as shewn in the figure. 



<«)• 



HI 



When the movable part of the key I iL 

is down, the condenser is in connexion ^ig- 127. 

with the cells and is charging; when the disc marked " insulate "^ 
on the key is pressed, the terminals of the condenser are in- 
sulated from each other ; when the disc marked " discharge " is 
pressed, the terminals are connected together through the 
galvanometer, and the condenser discharges itself. 

Owing to the passage of the discharge through the galva- 
nometer, the needle is deflected, and the extent of the first 
swing from the position of rest is proportional, within the degree 
of accuracy required for the present purpose, to the amount of 
electricity discharged through the galvanometer. 

Take several observations of the swings after the operation 
of charging has been performed in the least possible time. In 
order that the condenser may be properly discharged between 
each observation, allow the key to remain at the discharge 



LXXIII LEAKAGE AND ABSORPTION IN CONDENSERS 



351 



position for three minutes before again charging. The mean 
of these deflections may be taken to represent the charge of 
the condenser due to the applied electromotive force before 
leakage has had time to occur. They ought not to be less 
than 20 large scale-divisions. 

Now take several observations, charging the condenser for 
an instant only as before, but 
insulating for ten seconds, then 
discharging, and allowing three 
minutes before again charging. 

Repeat the observations for 
instantaneous charges followed by 
intervals of 30 seconds, 1, 2, 3, 4, 
and 5 minutes' insulation, and plot 
the results in the form of a curve, 
taking times as abscissae and de- 
flections as ordinates. 

Take logarithms of the de- pjg 128. 

flections, subtract each from the 

logarithm of the first deflection, divide by the differences in the 
time of insulation, and tabulate the results as follows : 

Condenser A. Leakage. 



















30,. 




tJy^^J r...^.. 






■-•n 


;"■ 


















20 


I 














.rge 


I 














10 


%N 








i 




Ni^ 


^-^ 


— 


Charged 
"Smts. 




1 ^^ 

1 1 






0^ 


1 


2 


. . .X 



' Time of 


Deflection 


Log. of 




Difference 


1 insolfttion 
Osec. 


on discharge 


deflection 


Difference 


Time 


30-9 


1-490 






10 „ 


23-3 


1-367 


•123 


•0123 


30 .. 


15-8 


1199 


•168 


•0084 


1 min. 


90 


•954 


•245 


•0082 


»7 


5-5 


•740 


•214 


•0036 


3 „ 


3-6 


•556 


•184 


•0031 


* » 


2-9 


•470 


•086 


•0014 


6 ,. 


2-4 


•380 


•090 


O015 



If the leakage at any instant were proportional to the 
charge in the condenser at that instant, the numbers in the 
last column would be equal. Their gradual diminution shews 
that the leakage increases more rapidly than the charge. 

In order to test whether the inequality is due to absorption 
of the charge by the dielectric of the condenser, the effect of 



352 



MAGNETISM AND ELECTRICITY 



LXXIII 



variation in the time of charge should now be found. The 
condenser should be charged each time for 10 seconds instead 
of an instant only, and the rest of the experiment carried out as 
previously. Then the charging should be continued for 30 
seconds and afterwards for 1 and for 5 minutes. 

Tables of results for each time of charge should be given 
and leakage curves drawn alongside the one drawn previously. 

If absorption by the dielectric occurs, the absorbed charge 
should make its reappearance after discharge on the insulated 
conductors of the condenser. To test this, if the condenser has 
a short circuiting key, insert it fur an instant to produce the 
first discharge without affecting the galvanometer, or if there 
is no key, arrange one in parallel with the condenser. Charge 
the condenser for 10 seconds, discharge for an instant through 
the short circuiting key and then insulate for 5 seconds. On 
now pressing the discharging key, the galvanometer will be 
deflected owing to the passage through it of the ''residual" 
discharge. 

Repeat the observations, allowing the condenser to remain 
insulated 10, 20, 40, and 60 seconds before taking the residual 
discharge. 

To find the influence of the time of charging on the 
magnitudes of the residual discharges, charge the condenser for 
10, 20, 40, and 60 seconds, insulate in each case for 5, 10, 20, 40, 
60 seconds and take the first residual discharges as before. 

Express the results by curves and in tabular form, as 
follows : 

Condenser A. First Residuals after charge for given time, 
discharge, and subsequent insulation for given time. 



Time of charge 


First Residuals after insulation for 


5 sees. 


10 sees. 


20 sees. 


40 sees. 


60 sees. 


5 seconds 
10 „ 
20 „ 
40 „ 
60 „ 


8-0 
10 
11 
13 
14 


9-0 
11 
13 
16 
18 


10 
12 
14 
17 

20 


11 
14 
17 
20 
23 


12 
17 
20 
23 
26 



LXXIII LEAKAGE AND ABSORPTION IN CONDENSERS 



353 















! 




' 1 


30 






1,11 

RMidiMljChwfM ; 












I 












20 














. 210.8ms. 1 










^ 


r^ 


^ 








10 


/ 


r^ 




_u 


—> 




ss. 1 




C^ 








' ■ 
















1 ' ! 






1 





10 


20 
Peri 


30 l40 ISO |60^ecofKt» 
>do/in.^lat•op. 1 1 










Fig. 


129. 









To test for successive residuals, charge again for 15 seconds, 
then discharge through the short circuiting key. Insulate for 
15 seconds, take the first residual, insulate for 15 seconds more 
and take the second residual, and so on till no more charge 
remains in the condenser. 

Tabulate as follows and draw a curve shewing the magni- 
tudes of the successive residuals as ordinates, and taking times 
of insulation since the first discharge, as abscissae. 



After insolation for 


Residual 
discharges 


15 seconds 
15 sees, more 
15 „ „ 
15 ., „ 


7-2 
6-4 
5-2 
41 



Take a second condenser not specially chosen to exhibit 
leakage and absorption, and test whether the leakage and 
absorption are small. 



8. p. 



SS 



SECTION LXXIV. 

COMPARISON AND USE OF CONDENSERS. 

Apparatus required : Condensers, discharge key, high 
resistance mirrcyr galvanometer, cells. 

It has been stated in the previous section that when a 
condenser is discharged through a galvanometer, the extent 
of the first swing from rest is proportional to the quantity of 
electricity discharged. If Q is this quantity, V the potential 
of the battery used to charge the condenser, and C the capacity 
of the condenser, then Q = CV. 

'J'he method of discharge may therefore be used either to 
compare the capacities of a number of condensers charged by 
the same battery and discharged through the galvanometer, or 
to compare the electromotive forces of a number of cells used 
to charge the same condenser. 

Apply the method, using a Leclanche cell and a high resist- 
ance galvanometer which swings without too much damping, 
to determine the capacities of the condensers provided, assuming 
that of the standard condenser to be given in microfarads. 
The condensers should be placed in circuit in turn, charged 
for an instant, then discharged, and the extent of the swing of 
the galvanometer needle observed. 

In order to see whether leakage has any effect on the 
observations, take discharges from each condenser after 10 
and 20 seconds' insulation, as well as instantaneous discharges. 



LXXIV COMPARISON AND USE OF CONDENSERS 355 

Record as follows: 

19 January, 1897. 
Nalder Galvanometer No. 5232. Leclanch^ Cell No. 12. 



Condenser 



Standard 

Swiss 

Paraffin No. I. 

II. 

III. 



Instan- 






Capacities 


taneous 


10 sees. 


20 8608. 


micro- 
farads 


23-7 


23-8 


23-7 


-33 


26-4 


25-2 


250 


•37 


27-2 


261 


25-4 


-38 



Remarks 



no leak 
small leak 



Select the condenser which shews least leakage, charge it 
in turn by means of a Leclanch^, a Daniell, a Clark and a 
storage cell. Take two discharges through the galvanometer 
with each cell, and assuming the electromotive force of a 
Clark to be 1'434 volts, calculate those of the other cells. 

Record as follows : 

Swiss Condenser. Nalder Galvanometer No. 5232. 



CeU 


Deflections 


Mean 


E.M.F. 


Leclanch6 
Daniell 
Storage Cell 
Clark 


34-9 35-0 
24-2 241 
48-1 47-5 
351 35-2 


34-95 
2415 
47-80 
3515 


1-42 voltiL 
•98 „ 
200 „ 
1-434 „ 



23—2 



SECTION LXXV. 

DETERMINATION OF THE CAPACITY OF 
A CONDENSER. 

Apparatus required : Condenser, high resistance galvano- 
meter, resistance coils, voltaic cell. 

When Q coulombs of electricity are discharged through a 
galvanometer the needle is deflected and swings through an 
angle a which, neglecting the damping, has been proved 
(page 342) to be given by the equation 

where HjG is the constant of the galvanometer. 

If the quantity Q is the charge of a condenser of capacity 
(7, due to an applied electromotive force E, then Q = GE and 
therefore 

^ I H T . a 

^ = E'G'^''''2' 

The damping may be taken into account by introducing the factor 

1 TT 

e^l'^ on the right-hand side. The quantity -^ . -^ may be deter- 
mined or eliminated, by observing the steady deflection of the 
galvanometer produced by the electromotive force E, acting 
through known resistances. 

Let a circuit be arranged as in Fig. 130, the cell being the 
same as that which has served to charge the condenser, and 
having its terminals connected to a resistance jR, which should 
be large compared to that of the cell, so that the difference 




LXXV DETERMINATION OF THE CAPACITY OF A CONDENSER 357 

of potential at the ends of R, may be taken to be equal to the 

electromotive force E. The terminals 

of the galvanometer are connected only 

to a small portion r of R, so that the 

difference of potential at the ends of the 

galvanometer circuit is ErjR. If the 

resistance of the galvanometer be p and 

the resistance p shewn in the figure be ^*^- ^^• 

not inserted, the current passing through the galvanometer will 

be Er/pR, and the permanent deflection will be given by the 

equation 

•'• GE~ pRtaind'' 
rTe^sinl 

irpR tan 6 ' 

The galvanometer resistance p is probably known from 
previous determinations, but may, if desired, be obtained ap- 
proximately or eliminated in the above expression, if a known 
resistance p' is inserted in the galvanometer circuit and the 
steady deflection d' observed. We have in that case 

Er H ^ ^ 

= ^ tan ^ ; 



(p-¥p)R G 

and combining this with the equation obtained for the case 
when p' is not inserted, we find 

p tan d' 
^""tan^-tan^* 

The condition most favourable for the determination of p 
by this method, is that in which p and p are equal so that 
the introduction of p approximately halves the deflection. 

The remarks made in Section LXXIL, page 34:^ as to 

6*''8m rt 
the best way of determining a apply to the present 



358 MAGNETISM AND ELECTRICITY LXXV 

section, and to vary the experiment the quantity X should 
in the present exercise be independently determined. 

Determine the capacity of the condenser provided, charging 
it by two Leclanche cells and discharging it through the galva- 
nometer. Take four observations of the first throw, two of 
which should be made with the battery reversed. 

Connect the galvanometer to the two Leclanche cells as 
shewn in Fig. 130, adding a commutator in the battery branch. 
Make R about 10,000 ohms, and r such that a deflection of 
about 1*2 times the discharge throw is obtained when p=0. 
Measure the deflection with the battery current in both direc- 
tions. Insert a resistance p' such that the deflection is about 
half what it was in the previous case, and again make two 
determinations of the deflection, reversing the battery current 
between the two observations. 

Measure the time of swing and the logarithmic decrement, 
following the instructions given on page 345. 

Record your observations as follows : 
3 October, 1900. 
Standard Condenser marked 1/3 microfarad. 
Galvanometer No. 5232. 

1. Discharge of Condenser after being charged by two 

Leclanche Cells. 
First deflection (mean of two deflections in 

each direction) = 165 cms. 

2. Measurement of steady Deflection. 

iR = 10000, r = 40. 
(a) p' = 0. Deflection (mean of one obser- 
vation in each direction) = 18*35 „ 
(6) p' = 10000. Deflection (do.) = 7-45 „ 
Hence galvanometer resistance 

7-4.^ 
^ = ^"""" 18-35- 745 -^««Q- 
A previous direct measurement of the galvanometer resist- 
ance had given p = 6570. The latter value, being more accurate, 
is adopted in the calculation. 



LXXV DETERMINATION OF THE CAPACITY OF A CONDENSER 359 



Time of 20 swings (a^ 


^erage of 


3 observati 


ions) = 749 seconds 


„ one swing 






= 3-745 ,. 


4. Determination of 


logarithi 


mic decrement. 




Successivt 


; deflections 






to right 


to left 




(1) 


1G16 


19-58 




(2) 


10-33 


12-37 




(3) 


680 


8-34 




(4) 


4-61 


5-41 




each number being the mean 


of three observations. 


Ratioof(3):(l) 


1 


= 


2-38 and 235 


„ ., (4) : (2] 


1 


= 


2-24 „ 2-29 


Mean Ratio 






2-31 


^_1 log 2-31 _ 


i X -3636 X 2-302 = 


•2092 


4 loge 








-t 




= 


1-105 



40x3-745 X 1105 x 1615 
10000 X 27r X 6880 x 1835 

= -337 X 10-« 

= *337 microfarad. 



APPENDIX. 



The following details, referring chiefly to the dimensions of the 
apparatus used in Owens College, will probably prove useful to 
teachers. 



SECTION 

IV. 



V. 



VI-VIII. 



XI. 



XI 11. 



Spirit Level. 

The ordinary 8 inch " brass adjusting level " is 
suitable. If the tube is not graduated, a paper 
scale may be gummed to it. 

Calibration Tube. 

A tube about 20 cms. long, of '7 cm. external 
and '07 cm. internal diameter, may be used. The 
tube and mercury should be quite dry. 

Balance and Density. 

A piece of quartz of 100-200 grams is a suitable 
body to weigh, since it can easily be kept clean, 
and therefore of constant weight. As quartz is 
unacted on by water it is also suitable for the 
density determinations. 

Moment of Inertia. 

The block should be suspended by means of a 
thin soft wire free from kinks. 

Compound Pendulum. 

A brass bar about a metre long, 2 cms. broad 
and -5 cm. thick, is suitable. The steel knife- 
edges are supported on glass plates. 



APPENDIX 



361 



SBCnON 

XIV. 



XV. 



XVI. 



XVII. 



XVIII. 



XIX. 



XX. 



Beams. 

Straight-grained beams about a metre long, 
1 -3 to 2 cms. broad and -7 to -9 cm. thick, should 
be used. 

Rigidity. 

The thinnest steel pianoforte wire (No. 30) and 
a wire about No. 25 are suitable. The ends of the 
wires are soldered to short lengths of brass wire of 
1*5 to 2 mms. diameter, one of which is tapped and 
screws into a hole in the axis of a cylindrical 
brass weight about 4 cms. diameter and 5 cms. 
long, and the other is clamped in the support. 

Viscosity. 

The tubes should have a length of 50 to 80 cms., 
the bore for water should be '5 to "8 mm., and for 
the calcium chloride solution 1*2 to 1*8 mms. A 
solution of density 11 5 to 1-20 (about 20 Vo) »» 
suitable. 

Surface Tension. 

The thin glass used tor covering microscope 
slides is suitable for the balance method. 

Expansion of a Solid. 

Tubes of about 1 cm. diameter and 60 or 
70 eras, long are convenient. 

Expansion of a Liquid. 

The graduated stem of a broken 0° to 100* C. 
mercury thermometer serves for the sttMu of the 
dilatoraeter. The bulb should have about 4 times 
the volume the bulb of the thermometer hod. 

Pressure Coefficient op a Gas. 

A little strong sulphuric acid should be placed 
in the bulb to keep the air dry. By warming 
the bulb slightly after a little mercury has been 
poured into the open tube, the volume of air 
enclosed may be reduced to the desired value. 



362 



APPENDIX 



SECTION 

XXI. 



XXII. 



XXIV. 



XXV. 
XXVI. 



XXVIII. 



XXIX. 



XXX. 



Expansion op a Gas. 

A tube of -l cm. bore and about 25 cms. long is 
suitable. The divisions etched on it may lt>e ren- 
dered more distinct by rubbing a little rouge into 
them. A strip of white enamelled glass should be 
placed between the tube and frame. 

Pressure and Boiling Point. 

A wide-necked pint bolt-head should be used 
for the boiling flask. A condenser 40 cms. long is 
necessary. 

Laws of Cooling. 

The calorimeter of copper -3 mm. thick is 6 5 cms. 
diameter and 8 cms. high. It stands on cork legs 
within a water-jacket, the inside diameter of which 
is 8*5, the outside 12*5, and the height 12 cms. 

Calorimetric Measurements. 

A piece of rubber 1x3x3 cms, is convenient. 
Specific Heat of Quartz. 

The steam heater is 40 cms. long, has an in- 
ternal diameter of 6*5 and an external of 9-5 cms. 
The internal cavity is closed at the top by a cork, 
and at the bottom by a metal door which can be 
moved aside so that the substance may be lowered 
into the calorimeter. 

Latent Heat op Steam. 

The condenser is of copper -3 mm. thick. The 
box is 4 X 3 X 1-5 cm., so that it can pass freely into 
the calorimeter. 

Heat op Solution. 

A calorimeter of thin copper 3 5 cms. diameter 
and 4 cms. high is used to hold the salt. 

Mechanical Equivalent op Heat. 

The apparatus used is that of Puluj, with the 
arrangement for measuring the frictional couple 
improved. To diminish the correction for cooling, 
the cross-section of the float has been increased to 
six or seven times that given on p. 145. 



APPENDIX 



363 



SECTIOS 

XXXI. 



XXXIIL 



XXXV. 



XXXVI. 



XXXVII. 



XXXVIII. 



XXXIX 



XLI. 



XLII. 



FrtEQUENCY BY THE SyREN. 

It is convenient to apply part of the pressure 
on the top of the bellows by hand, as this enables 
the note of the syren to be readily varied by a 
slight change of the force used. 

LiSSAJOUS* FlOURES. 

Large forks of frequencies 128 and 256 are 
suitable. 

Sextant. 

The objects viewed should be at a considerable 
distance from the observer, and their angular 
distance apart should not be greater than about 
20". 

Curvature of Lenses. 

The optic bench is 1 metre long and 14 cms. wide. 
The lens is an ordinary reading lens of II cms. 
diameter. 

Total Reflection. 

The cube has an edge of 4*5 cms., and the slit is 
22 cms. above tiie table. 

Magnifying powers. 

A board painted white, with a black .scale 
painted on it with divisions about 8 cms. apart, and 
each fifth numlxired, makes a suitable object for the 
first exercise. 
Use of Spectroscope. 

The salts are contained in 2 inch corked sample 
bottles placed in holes in a block of wood 20 cms. 
by 7 cms. The crayons are the ordinary coloured 
crayons sold in sixpenny boxes. 
The Spectrometer and its Adjustments. 

The model of the vernier is about '^0 cms. long. 
The graduations are on paper gunimc<l to wooden 
blocks and varnished. 
Refra<tivk Inoe.x of a Solid. 

The small piece of mirror is provided to enable 
light to be reflected down on to the verniers when 
the readings are l)eing taken. 



364 



APPENDIX 



SECTION 

XLIV. 



XLVI. 



XLVII. 



XLVIII. 



XLIX. 



Photometry. 



The standard sperm candle weighs J 



of a 



pound. 



LII. 



LIII. 



Diffraction Grating. 

One of Thorp's replicas answers the purpose 
admirably. 

Rotation of Plane of Polarisation. I. 

The tubes containing the sugar solution should 
be of glass or be glass-lined, to prevent the acid 
used in inverting the sugar acting on them. 

Rotation of Plane of Polarisation. II. 

There are other means of producing the half 
shadow field, as effective as the plate of quartz 
described on p. 227. 

Magnetic Fields. 

The horizontal graduated scales are about 3 mms. 
thick and 80 or 100 cms. long. A deflecting magnet 
about 6 X -5 X '6 cms. is suitable. 

Magnetic Survey. 

The apparatus consists of a circular scale of 
15 — 20 cms. diameter, over which a magnetic needle 
4 cms. long with a cross section of 04 — '06 sq. 
cm., is suspended by a line untwisted silk fibre 
about 20 cms. long. The needle should have a light 
aluminium pointer attached to it. The scale and 
needle are enclosed in a box open at the top. 

Comparison of Current Meters. 

The resistance coils are of No. 16 platinoid wire, 
and are arranged parallel to each other between 
terminals which may be connected together in any 
desired way by means of strips of copper. 

Application op Ohm's Law, 

The wires are stretched along graduated metre 
scales screwed to rods of wood 1 '5 cms. thick and 
6 cms. broad. 



APPENDIX 



365 



SECTION 

LIV. 



LV. 



LVI. 



LVIII. 



UX. 



LX. 



LXII. 



LXIIl. 



LXIV. 



Arrangement of Cells. 

The outer jars of the cells are 13 cms. high and 
10 cms. diameter outside, and the porous pots 
12 cms. high and 5*5 cms. diameter. 

Water Voltameter. 

The voltameter of Fig. 105 is due to Kohlrausch 
and may be obtained from Messrs Hartmann and 
Braun, Bockenheim near Frankfort o. M. 

Copper Voltameter. 

The copper plates used are about 6 cms. square, 
and each has a lug about 3 cms. long, which fits 
into and makes good electrical contact with a 
spring clip. 

Measurement of Resistance. 

Coils of alK)ut 4, 20, 100, 500, and 2000 ohms 
give sufficient practice to the student. 

High Resistance. 

Strips of insulating material clamped between 
binding screws answer for the last portion of the 
exercise. 

Low Resistance. 

A coil of about 30 cms. of No. 14 platinoid wire, 
with connexions made to it at several points of its 
length, is convenient. 

Resistance of a Cell. 

A condenser of J to 1 microfarad capacity u 
suitable. 

Carey Foster's Method. 

The two equal coils may conveniently be made 
of bare platinoid or manganin wire wound on glass 
and kept in the same bath of petroleum. 

Temperature Change of Resistance. 

About 4 metres of No. 38 copper, and 2 metres 
of No. 29 platinoid or manganin wire wound on 
strips of insulating fibre, form suitable coils. 



366 



APPENDIX 



SECTION 

LXV. 



LXVI. 



LXVIII. 



LXX. 



LXXI. 



LXXII. 



LXXIII. 



Resistance of Electrolytes. 

A tube about 16 cms. long and 4 -5 cms. internal 
diameter enables all the solutions to be tested by 
means of one Post OflSce box of 1 1 ,000 ohms. 

Clark Cell. 

The vertical limbs of the H tube are 8 cms. long 
and 1 -6 cms. diameter. 

Potentiometer Method of Measuring Currents. 
The wire used for the standard low resistance 
should be capable of carrying the current without 
the increase of temperature produced altering its 
resistance appreciably. 

Equivalent of Heat by the Electrical Method. 

The heating coil consists of 90 cms. of No. 14 
platinoid wire. 

Induction of Electric Currents. 

The 3 inner coils each consist of one layer of 50 
turns of No. 20 copper wire, and are wound in 
parallel. The 3 outer coils are similar but have 40 
turns. Short lengths of platinoid wire are inter- 
posed between the ends of the coils and the terminals, 
so that whether one coil, or two in series, or three 
in series are in circuit the resistance is the same. 

Inductance of a Coil. 

The coil consists of 575 turns of No. 20 copper 
wire, the mean radius of a turn being 25-3 cms. 

Leakage and Absorption of Condensers. 

The dielectric of the condenser used consists 
of paraffined paper, one side of which is shellac 
varnished. 



INDEX. 



Accurate weighing, 40 
Alcohol, refractive index of, 210 
Angles, optical methods of measuring, 
154 

Balance, 28 

,, sensibility of, 33 
Bifilar suspension, 71 
Biprism, 216 
Boiling-point depending on pressure, 

112 
Buoyancy correction, 41, 48 

Calibration of graduated tube, 22 

„ spirit level, 17 
Capacity of condensers, 357 
Clark cell, 319 
Compound pendulum, 75 
Condensers, 350 

„ capacity of, 357 

„ comparison of, 354 

„ leakage of, 351 

„ residual charge, 352 

Cooling correction in calorimetry, 
126 
„ laws of, 119 
„ Newton's law, 123 
Copper voltameter, 267 
Current meters, comparison of, 248 
,, ,, standardisation of, 

267 
Currents measured by potentiometer, 

827 
Cniratures of lenses, 157 

Density of a liquid, 62 

„ ,, quartz, 65, 58 

„ ,, a solid, 53 
Dew point, 116 
Diffraction grating, 220 
Discarding of anneoessary deoinuds, 

181 

Earth's horizontal force, 284 
Bleeiro^ynamometer, 246 
Electrolytes, resistance of, 816 



Electromotive forces, comparison of» 

323 
Equivalent simple pendulum, 76 
Equivalent simple pendulum, deter- 
mination of, 79 
Error, fractional, 6 

„ probable, 3 
Errors of observation, 1 
Expansion of a gas at constant prea> 
sure, 108 
„ M liquid, 100 

solid, 97 

Foster's method of comparing resis- 
tances, 307 
Frequency of tuning fork, 146 

Galvanometer, adjustment of, 272 

,, constant by water elec- 

trolysis, 265 
„ dead beat, 282 

,, how to increase sen- 

sitiveness, 279 
,, resistance of, 800 

,, tangent adjustment of, 

245 
Gauss's eyepiece, 200 

„ method of weighing, 45 

Heat of solution, 140 
Humidity, relative, 116 
Hydrometer, 65 
Hygrometer, Daniell's, 116 

India-rubber, specific heat of, 128 
Inductance, determination of, 840 
Induction, 835 
Inertia, moments of, 67, 71 
Interference of light, 216 

Kelvin current balance, 271 
Kundt's method of measuring velocity 
of sound, 149 

Latent beat of steam, 187 
„ ,, water, 185 



368 



INDEX 



Laurent's polarimeter, 229 

Least squares, method of, 7 

Length, measurement of, 9 

Lenses, curvatures and powers, 167 

Level, calibration of, 17 

Lissajous' figures, 151 

Lodge's modification of Mance*iB 

method, 304 
Logarithmic decrement, 283 

Magnetic fields, 232 

,, intensity, 238 
Magnifying power, 176 
Mance's method, 304 
Mechanical equivalent of heat by 

electrical method, 332 
Mechanical equivalent of heat by 

friction, 142 
Modulus of rigidity, 86 

Young's, 72 
Mohr's balance, 64 
Moments of inertia, 57, 71 

Newton's law of cooling, 123 

Ohm's law, verification of, 253 

Personal equation, 8 

Photometry, 213 

Platinum thermometers, 313 

Polarisation, rotation of plane, 224, 

226 
Post Office box, 286 
Potentiometer, 325 

,, method of measuring 

currents, 327 
Pressure, effect on boiling point, 112 

,, coefficient of gas, 104 
Prism, determination of angle, 205 
Puluj's friction cones, 142 

Quartz, specific heat of, 132 

Refractive index, 173 

„ „ of liquids, 209 

of solids, 202 
„ power, 212 

Resistance of a cell, 304 

„ of electrolytes, 315 ; brass, 

expansion of, 99 
„ of a galvanometer, 300 



Resistance, high, measurement of, 293 

,, low, measurement of, 298 

,, measurement of, 285, 307 

„ temperature coefficient of, 

310 

Resistivity, measurement of, 292 

Eider of balance, 33 

Rigidity, modulus of, 86 



Saturation, fractional, 116 

Sensibility of balance, 33 

Sextant, 162 

,, index error, 165 

Specific gravity, 53 

„ „ flask, 57 

,, heat of india-rubber, 
,, „ quartz, 132 

Spectra, mapping of, 184 

Spectrometer, adjustment of. 

Spectroscope, adjustment of, 

,, reduction to 

lengths, 187 

Steam, latent heat of, 137 

Subdivision by eye, 9 

Surface tension, 93 

Syren, 147 



128 



194 
181 
wave 



Temperature coefficient of electrical 

resistance, 310 
Temperature coefficient of expansion 

of gas, 108 
Thermo-electricity, 329 
Time, intervals of, 12 
Total reflection, 173 
Tube, calibration of, 22 

Velocity of sound, measurement of, 

149 
Viscosity, 89 

Voltaic cells, arrangement of, 258 
Voltameter, water, 262 
,, copper, 267 

Water, expansion of, 102 

,, latent heat of, 135 

„ voltameter, 262 
Wave length, measurement of, 220 
Wet and dry bulb thermometer, 118 

Young's modulus, 72 



CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. 



UNIVERSITY OF TORONTO 
LIBRARY 




Acme Library Card Pockvt 

Inder P»t. " Ref. lodet File." 
Made hj LIBRABT BIJILEAU